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Matching statistic: St001875
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Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,5,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,5,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,4,5,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,5,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,4,1,5,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,5,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,5,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3,5] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,2,5,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,5,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[5,3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,3,4,5,2,6] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,4,5,6,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,4,6,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,5,2,4,6] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,5,2,6,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,5,6,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,6,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,6,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,4,2,5,3,6] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,4,2,5,6,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,4,2,6,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,4,5,2,3,6] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001615
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,5,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,5,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,4,5,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,5,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,1,5,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,5,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,5,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,1,2,3,5] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,2,5,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,5,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,4,5,2,6] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,4,5,6,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,4,6,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,5,2,4,6] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,5,2,6,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,5,6,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,6,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,6,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,5,3,6] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,5,6,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,6,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,5,2,3,6] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
Description
The number of join prime elements of a lattice.
An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001617
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,5,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,5,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,4,5,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,5,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,1,5,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,5,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,5,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,1,2,3,5] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,2,5,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,5,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,4,5,2,6] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,4,5,6,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,4,6,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,5,2,4,6] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,5,2,6,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,5,6,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,6,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,6,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,5,3,6] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,5,6,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,6,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,5,2,3,6] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
Description
The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying
$$
v(a\vee b) + v(a\wedge b) = v(a) + v(b).
$$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]].
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Matching statistic: St001622
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,5,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,5,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,4,5,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,5,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,1,5,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,5,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,5,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,1,2,3,5] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,2,5,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,5,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,4,5,2,6] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,4,5,6,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,4,6,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,5,2,4,6] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,5,2,6,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,5,6,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,6,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,6,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,5,3,6] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,5,6,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,6,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,5,2,3,6] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St001616
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 1
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 3 - 1
[1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 3 - 1
[4,1,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 3 - 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 4 - 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 1
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 1
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 1
[2,3,4,1,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[2,3,4,5,1] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[2,3,5,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 1
[2,4,1,3,5] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 3 - 1
[2,4,1,5,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 3 - 1
[2,4,5,1,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 3 - 1
[2,5,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 1
[2,5,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 3 - 1
[3,1,4,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 1
[3,1,4,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 1
[3,1,5,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 1
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 1
[3,4,1,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 1
[3,4,5,1,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 1
[3,5,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 1
[3,5,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 1
[4,1,2,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 1
[4,1,2,5,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[4,1,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[4,5,1,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[5,1,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 1
[5,2,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 1
[5,2,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 3 - 1
[5,3,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 1
[5,3,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 1
[5,4,1,2,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 5 - 1
[1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 1
[1,2,3,6,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 1
[1,2,6,3,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 1
[1,3,4,5,2,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 1
[1,3,4,5,6,2] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 1
[1,3,4,6,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 1
[1,3,5,2,4,6] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 1
[1,3,5,2,6,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 1
[1,3,5,6,2,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 1
[1,3,6,4,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 1
[1,3,6,5,2,4] => [[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 1
[1,4,2,5,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 1
[1,4,2,5,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 1
[1,4,2,6,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 1
[1,4,5,2,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 1
[1,4,5,2,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 1
[1,4,5,6,2,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 1
[1,4,6,2,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 1
[1,4,6,5,2,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 1
[1,5,2,3,4,6] => [[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3 - 1
[1,5,2,3,6,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 1
[1,5,2,6,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 1
[1,5,6,2,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 1
[2,4,1,3,6,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[2,4,1,6,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[2,4,6,1,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[4,2,5,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,2,5,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,2,5,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,2,6,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,5,2,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,5,2,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,5,2,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,5,6,2,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,6,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,6,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[5,2,3,6,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 3 - 1
[5,2,6,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 3 - 1
[5,6,2,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 3 - 1
[6,4,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 3 - 1
[4,2,5,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,2,5,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,2,5,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,2,5,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,2,5,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,5,2,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,5,2,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,5,2,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,5,2,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,5,2,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,5,7,2,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[4,5,7,2,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[5,2,4,1,3,7,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[5,2,4,1,7,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[5,2,4,7,1,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[6,2,4,1,3,7,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[6,2,4,1,7,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[6,2,4,7,1,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[6,3,7,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[6,4,2,7,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[6,4,7,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[6,4,7,5,2,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[6,7,3,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
[6,7,4,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 3 - 1
Description
The number of neutral elements in a lattice.
An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St001613
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 2
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 3 - 2
[1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 3 - 2
[4,1,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 3 - 2
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 4 - 2
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[2,3,4,1,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 2
[2,3,4,5,1] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 2
[2,3,5,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
[2,4,1,3,5] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[2,4,1,5,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[2,4,5,1,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[2,5,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
[2,5,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[3,1,4,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,1,4,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,1,5,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,4,1,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,4,5,1,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,5,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[3,5,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[4,1,2,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 2
[4,1,2,5,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
[4,1,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
[4,5,1,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
[5,1,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[5,2,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
[5,2,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[5,3,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[5,3,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[5,4,1,2,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 2
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 5 - 2
[1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 2
[1,2,3,6,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 2
[1,2,6,3,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 2
[1,3,4,5,2,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,3,4,5,6,2] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,3,4,6,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 2
[1,3,5,2,4,6] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,3,5,2,6,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,3,5,6,2,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,3,6,4,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 2
[1,3,6,5,2,4] => [[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,4,2,5,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,2,5,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,2,6,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 2
[1,4,5,2,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,5,2,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,5,6,2,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,6,2,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 2
[1,4,6,5,2,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 2
[1,5,2,3,4,6] => [[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3 - 2
[1,5,2,3,6,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,5,2,6,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,5,6,2,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 2
[2,4,1,3,6,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 3 - 2
[2,4,1,6,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 3 - 2
[2,4,6,1,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 3 - 2
[4,2,5,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,5,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,5,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,6,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,2,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,2,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,2,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,6,2,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,6,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,6,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[5,2,3,6,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 3 - 2
[5,2,6,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 3 - 2
[5,6,2,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 3 - 2
[6,4,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,5,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,7,2,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,7,2,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[5,2,4,1,3,7,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[5,2,4,1,7,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[5,2,4,7,1,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,2,4,1,3,7,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,2,4,1,7,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,2,4,7,1,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,3,7,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,4,2,7,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,4,7,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,4,7,5,2,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,7,3,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,7,4,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001719
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 2
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 3 - 2
[1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 3 - 2
[4,1,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 3 - 2
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 4 - 2
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[2,3,4,1,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 2
[2,3,4,5,1] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 2
[2,3,5,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
[2,4,1,3,5] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[2,4,1,5,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[2,4,5,1,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[2,5,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
[2,5,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[3,1,4,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,1,4,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,1,5,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,4,1,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,4,5,1,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,5,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[3,5,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[4,1,2,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 2
[4,1,2,5,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
[4,1,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
[4,5,1,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
[5,1,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[5,2,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
[5,2,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[5,3,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[5,3,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[5,4,1,2,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 2
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 5 - 2
[1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 2
[1,2,3,6,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 2
[1,2,6,3,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 2
[1,3,4,5,2,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,3,4,5,6,2] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,3,4,6,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 2
[1,3,5,2,4,6] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,3,5,2,6,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,3,5,6,2,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,3,6,4,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 2
[1,3,6,5,2,4] => [[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,4,2,5,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,2,5,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,2,6,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 2
[1,4,5,2,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,5,2,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,5,6,2,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,6,2,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 2
[1,4,6,5,2,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 2
[1,5,2,3,4,6] => [[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3 - 2
[1,5,2,3,6,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,5,2,6,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,5,6,2,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 2
[2,4,1,3,6,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 3 - 2
[2,4,1,6,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 3 - 2
[2,4,6,1,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 3 - 2
[4,2,5,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,5,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,5,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,6,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,2,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,2,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,2,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,6,2,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,6,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,6,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[5,2,3,6,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 3 - 2
[5,2,6,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 3 - 2
[5,6,2,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 3 - 2
[6,4,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,5,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,7,2,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,7,2,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[5,2,4,1,3,7,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[5,2,4,1,7,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[5,2,4,7,1,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,2,4,1,3,7,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,2,4,1,7,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,2,4,7,1,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,3,7,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,4,2,7,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,4,7,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,4,7,5,2,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,7,3,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,7,4,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001881
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 2
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 3 - 2
[1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 3 - 2
[4,1,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 3 - 2
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 4 - 2
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[2,3,4,1,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 2
[2,3,4,5,1] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 2
[2,3,5,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
[2,4,1,3,5] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[2,4,1,5,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[2,4,5,1,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[2,5,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
[2,5,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[3,1,4,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,1,4,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,1,5,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,4,1,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,4,5,1,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
[3,5,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[3,5,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[4,1,2,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 2
[4,1,2,5,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
[4,1,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
[4,5,1,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
[5,1,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 2
[5,2,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
[5,2,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 3 - 2
[5,3,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[5,3,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 2
[5,4,1,2,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 2
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 5 - 2
[1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 2
[1,2,3,6,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 2
[1,2,6,3,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 2
[1,3,4,5,2,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,3,4,5,6,2] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,3,4,6,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 2
[1,3,5,2,4,6] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,3,5,2,6,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,3,5,6,2,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,3,6,4,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 2
[1,3,6,5,2,4] => [[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 2
[1,4,2,5,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,2,5,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,2,6,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 2
[1,4,5,2,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,5,2,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,5,6,2,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 2
[1,4,6,2,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 2
[1,4,6,5,2,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 2
[1,5,2,3,4,6] => [[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3 - 2
[1,5,2,3,6,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,5,2,6,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 2
[1,5,6,2,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 2
[2,4,1,3,6,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 3 - 2
[2,4,1,6,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 3 - 2
[2,4,6,1,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 3 - 2
[4,2,5,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,5,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,5,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,6,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,2,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,2,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,2,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,5,6,2,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,6,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,6,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[5,2,3,6,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 3 - 2
[5,2,6,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 3 - 2
[5,6,2,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 3 - 2
[6,4,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 3 - 2
[4,2,5,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,2,5,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,2,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,7,2,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[4,5,7,2,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[5,2,4,1,3,7,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[5,2,4,1,7,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[5,2,4,7,1,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,2,4,1,3,7,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,2,4,1,7,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,2,4,7,1,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,3,7,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,4,2,7,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,4,7,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,4,7,5,2,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,7,3,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
[6,7,4,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 3 - 2
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Matching statistic: St001846
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 - 3
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 0 = 3 - 3
[1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 0 = 3 - 3
[4,1,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 0 = 3 - 3
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 4 - 3
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 3
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 3
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 3
[2,3,4,1,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 3
[2,3,4,5,1] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 3
[2,3,5,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 3
[2,4,1,3,5] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 0 = 3 - 3
[2,4,1,5,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 0 = 3 - 3
[2,4,5,1,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 0 = 3 - 3
[2,5,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 3
[2,5,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 0 = 3 - 3
[3,1,4,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 3
[3,1,4,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 3
[3,1,5,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 3
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 3
[3,4,1,5,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 3
[3,4,5,1,2] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 3
[3,5,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 3
[3,5,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 3
[4,1,2,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 3
[4,1,2,5,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 3
[4,1,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 3
[4,5,1,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 3
[5,1,2,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 4 - 3
[5,2,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 3
[5,2,4,1,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 0 = 3 - 3
[5,3,1,4,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 3
[5,3,4,1,2] => [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3 - 3
[5,4,1,2,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 3
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 5 - 3
[1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 3
[1,2,3,6,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 3
[1,2,6,3,4,5] => [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 5 - 3
[1,3,4,5,2,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 3
[1,3,4,5,6,2] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? = 3 - 3
[1,3,4,6,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 3
[1,3,5,2,4,6] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 3
[1,3,5,2,6,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 3
[1,3,5,6,2,4] => [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 3
[1,3,6,4,5,2] => [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 3 - 3
[1,3,6,5,2,4] => [[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 3 - 3
[1,4,2,5,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 3
[1,4,2,5,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 3
[1,4,2,6,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 3
[1,4,5,2,3,6] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 3
[1,4,5,2,6,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 3
[1,4,5,6,2,3] => [[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 3 - 3
[1,4,6,2,5,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 3
[1,4,6,5,2,3] => [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 3 - 3
[1,5,2,3,4,6] => [[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 3 - 3
[1,5,2,3,6,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 3
[1,5,2,6,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 3
[1,5,6,2,3,4] => [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 3 - 3
[2,4,1,3,6,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[2,4,1,6,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[2,4,6,1,3,5] => [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[4,2,5,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,2,5,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,2,5,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,2,6,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,5,2,3,1,6] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,5,2,3,6,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,5,2,6,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,5,6,2,3,1] => [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,6,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,6,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[5,2,3,6,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0 = 3 - 3
[5,2,6,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0 = 3 - 3
[5,6,2,3,4,1] => [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0 = 3 - 3
[6,4,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 3 - 3
[4,2,5,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,2,5,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,2,5,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,2,5,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,2,5,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,5,2,3,1,7,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,5,2,3,7,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,5,2,3,7,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,5,2,7,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,5,2,7,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,5,7,2,3,1,6] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[4,5,7,2,3,6,1] => [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[5,2,4,1,3,7,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[5,2,4,1,7,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[5,2,4,7,1,3,6] => [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[6,2,4,1,3,7,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[6,2,4,1,7,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[6,2,4,7,1,3,5] => [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[6,3,7,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[6,4,2,7,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[6,4,7,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[6,4,7,5,2,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[6,7,3,5,2,4,1] => [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
[6,7,4,2,5,3,1] => [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 0 = 3 - 3
Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
Matching statistic: St001618
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001618: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001618: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 3 - 1
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 3 - 1
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 1
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
=> ? = 4 - 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 4 - 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 4 - 1
[1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 4 - 1
[2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 3 - 1
[2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 3 - 1
[2,3,5,4,1] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 3 - 1
[2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3 - 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 3 - 1
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
[2,5,3,4,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 1
[2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 - 1
[3,1,4,2,5] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3 - 1
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 1
[3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 3 - 1
[3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 1
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
[3,4,5,1,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 1
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 - 1
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[4,1,2,3,5] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 3 - 1
[4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 1
[4,1,5,2,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
[4,5,1,2,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 1
[5,1,2,3,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 4 - 1
[5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[5,3,1,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[5,3,4,1,2] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[5,4,1,2,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(1,82),(1,83),(1,84),(1,85),(2,18),(2,19),(2,25),(2,30),(2,31),(2,37),(2,78),(2,79),(2,81),(2,83),(3,16),(3,17),(3,24),(3,28),(3,29),(3,36),(3,76),(3,77),(3,81),(3,82),(4,21),(4,23),(4,27),(4,33),(4,35),(4,39),(4,77),(4,79),(4,80),(4,85),(5,20),(5,22),(5,26),(5,32),(5,34),(5,38),(5,76),(5,78),(5,80),(5,84),(6,22),(6,23),(6,24),(6,46),(6,47),(6,54),(6,83),(6,86),(6,87),(6,91),(7,20),(7,21),(7,25),(7,48),(7,49),(7,55),(7,82),(7,88),(7,89),(7,91),(8,17),(8,19),(8,26),(8,50),(8,52),(8,56),(8,85),(8,86),(8,88),(8,90),(9,16),(9,18),(9,27),(9,51),(9,53),(9,57),(9,84),(9,87),(9,89),(9,90),(10,28),(10,32),(10,43),(10,48),(10,51),(10,59),(10,79),(10,86),(10,92),(10,94),(11,29),(11,33),(11,42),(11,49),(11,50),(11,60),(11,78),(11,87),(11,92),(11,95),(12,30),(12,34),(12,41),(12,46),(12,53),(12,60),(12,77),(12,88),(12,93),(12,94),(13,31),(13,35),(13,40),(13,47),(13,52),(13,59),(13,76),(13,89),(13,93),(13,95),(14,38),(14,39),(14,45),(14,56),(14,57),(14,58),(14,81),(14,91),(14,94),(14,95),(15,36),(15,37),(15,44),(15,54),(15,55),(15,58),(15,80),(15,90),(15,92),(15,93),(16,61),(16,107),(16,112),(16,126),(16,134),(16,170),(17,62),(17,106),(17,113),(17,127),(17,134),(17,171),(18,63),(18,109),(18,112),(18,129),(18,135),(18,172),(19,64),(19,108),(19,113),(19,128),(19,135),(19,173),(20,65),(20,104),(20,110),(20,132),(20,136),(20,170),(21,66),(21,105),(21,110),(21,133),(21,137),(21,171),(22,67),(22,102),(22,111),(22,130),(22,136),(22,172),(23,68),(23,103),(23,111),(23,131),(23,137),(23,173),(24,69),(24,102),(24,103),(24,126),(24,127),(24,175),(25,70),(25,104),(25,105),(25,128),(25,129),(25,175),(26,71),(26,106),(26,108),(26,130),(26,132),(26,174),(27,72),(27,107),(27,109),(27,131),(27,133),(27,174),(28,61),(28,96),(28,114),(28,127),(28,138),(28,169),(29,62),(29,97),(29,115),(29,126),(29,138),(29,168),(30,63),(30,98),(30,117),(30,128),(30,139),(30,169),(31,64),(31,99),(31,116),(31,129),(31,139),(31,168),(32,65),(32,96),(32,118),(32,130),(32,140),(32,167),(33,66),(33,97),(33,119),(33,131),(33,141),(33,167),(34,67),(34,98),(34,120),(34,132),(34,140),(34,166),(35,68),(35,99),(35,121),(35,133),(35,141),(35,166),(36,69),(36,100),(36,122),(36,134),(36,138),(36,166),(37,70),(37,100),(37,123),(37,135),(37,139),(37,167),(38,71),(38,101),(38,124),(38,136),(38,140),(38,168),(39,72),(39,101),(39,125),(39,137),(39,141),(39,169),(40,73),(40,116),(40,121),(40,142),(40,144),(40,170),(41,74),(41,117),(41,120),(41,142),(41,145),(41,171),(42,74),(42,115),(42,119),(42,143),(42,144),(42,172),(43,73),(43,114),(43,118),(43,143),(43,145),(43,173),(44,75),(44,122),(44,123),(44,142),(44,143),(44,174),(45,75),(45,124),(45,125),(45,144),(45,145),(45,175),(46,67),(46,103),(46,117),(46,148),(46,152),(46,178),(47,68),(47,102),(47,116),(47,149),(47,152),(47,179),(48,65),(48,105),(48,114),(48,146),(48,153),(48,178),(49,66),(49,104),(49,115),(49,147),(49,153),(49,179),(50,62),(50,108),(50,119),(50,147),(50,154),(50,176),(51,61),(51,109),(51,118),(51,146),(51,155),(51,176),(52,64),(52,106),(52,121),(52,149),(52,154),(52,177),(53,63),(53,107),(53,120),(53,148),(53,155),(53,177),(54,69),(54,111),(54,123),(54,150),(54,152),(54,176),(55,70),(55,110),(55,122),(55,150),(55,153),(55,177),(56,71),(56,113),(56,125),(56,151),(56,154),(56,178),(57,72),(57,112),(57,124),(57,151),(57,155),(57,179),(58,75),(58,100),(58,101),(58,150),(58,151),(58,180),(59,73),(59,96),(59,99),(59,146),(59,149),(59,180),(60,74),(60,97),(60,98),(60,147),(60,148),(60,180),(61,181),(61,189),(61,190),(62,181),(62,188),(62,191),(63,182),(63,189),(63,192),(64,182),(64,188),(64,193),(65,183),(65,187),(65,190),(66,184),(66,187),(66,191),(67,183),(67,186),(67,192),(68,184),(68,186),(68,193),(69,181),(69,186),(69,194),(70,182),(70,187),(70,194),(71,183),(71,188),(71,195),(72,184),(72,189),(72,195),(73,185),(73,190),(73,193),(74,185),(74,191),(74,192),(75,185),(75,194),(75,195),(76,96),(76,102),(76,106),(76,166),(76,168),(76,170),(77,97),(77,103),(77,107),(77,166),(77,169),(77,171),(78,98),(78,104),(78,108),(78,167),(78,168),(78,172),(79,99),(79,105),(79,109),(79,167),(79,169),(79,173),(80,101),(80,110),(80,111),(80,166),(80,167),(80,174),(81,100),(81,112),(81,113),(81,168),(81,169),(81,175),(82,114),(82,115),(82,122),(82,170),(82,171),(82,175),(83,116),(83,117),(83,123),(83,172),(83,173),(83,175),(84,118),(84,120),(84,124),(84,170),(84,172),(84,174),(85,119),(85,121),(85,125),(85,171),(85,173),(85,174),(86,127),(86,130),(86,149),(86,173),(86,176),(86,178),(87,126),(87,131),(87,148),(87,172),(87,176),(87,179),(88,128),(88,132),(88,147),(88,171),(88,177),(88,178),(89,129),(89,133),(89,146),(89,170),(89,177),(89,179),(90,134),(90,135),(90,151),(90,174),(90,176),(90,177),(91,136),(91,137),(91,150),(91,175),(91,178),(91,179),(92,138),(92,143),(92,153),(92,167),(92,176),(92,180),(93,139),(93,142),(93,152),(93,166),(93,177),(93,180),(94,140),(94,145),(94,155),(94,169),(94,178),(94,180),(95,141),(95,144),(95,154),(95,168),(95,179),(95,180),(96,156),(96,190),(96,197),(97,157),(97,191),(97,197),(98,158),(98,192),(98,197),(99,159),(99,193),(99,197),(100,160),(100,194),(100,197),(101,161),(101,195),(101,197),(102,156),(102,186),(102,200),(103,157),(103,186),(103,201),(104,158),(104,187),(104,200),(105,159),(105,187),(105,201),(106,156),(106,188),(106,198),(107,157),(107,189),(107,198),(108,158),(108,188),(108,199),(109,159),(109,189),(109,199),(110,161),(110,187),(110,198),(111,161),(111,186),(111,199),(112,160),(112,189),(112,200),(113,160),(113,188),(113,201),(114,162),(114,190),(114,201),(115,162),(115,191),(115,200),(116,163),(116,193),(116,200),(117,163),(117,192),(117,201),(118,164),(118,190),(118,199),(119,165),(119,191),(119,199),(120,164),(120,192),(120,198),(121,165),(121,193),(121,198),(122,162),(122,194),(122,198),(123,163),(123,194),(123,199),(124,164),(124,195),(124,200),(125,165),(125,195),(125,201),(126,157),(126,181),(126,200),(127,156),(127,181),(127,201),(128,158),(128,182),(128,201),(129,159),(129,182),(129,200),(130,156),(130,183),(130,199),(131,157),(131,184),(131,199),(132,158),(132,183),(132,198),(133,159),(133,184),(133,198),(134,160),(134,181),(134,198),(135,160),(135,182),(135,199),(136,161),(136,183),(136,200),(137,161),(137,184),(137,201),(138,162),(138,181),(138,197),(139,163),(139,182),(139,197),(140,164),(140,183),(140,197),(141,165),(141,184),(141,197),(142,163),(142,185),(142,198),(143,162),(143,185),(143,199),(144,165),(144,185),(144,200),(145,164),(145,185),(145,201),(146,159),(146,190),(146,196),(147,158),(147,191),(147,196),(148,157),(148,192),(148,196),(149,156),(149,193),(149,196),(150,161),(150,194),(150,196),(151,160),(151,195),(151,196),(152,163),(152,186),(152,196),(153,162),(153,187),(153,196),(154,165),(154,188),(154,196),(155,164),(155,189),(155,196),(156,202),(157,202),(158,202),(159,202),(160,202),(161,202),(162,202),(163,202),(164,202),(165,202),(166,186),(166,197),(166,198),(167,187),(167,197),(167,199),(168,188),(168,197),(168,200),(169,189),(169,197),(169,201),(170,190),(170,198),(170,200),(171,191),(171,198),(171,201),(172,192),(172,199),(172,200),(173,193),(173,199),(173,201),(174,195),(174,198),(174,199),(175,194),(175,200),(175,201),(176,181),(176,196),(176,199),(177,182),(177,196),(177,198),(178,183),(178,196),(178,201),(179,184),(179,196),(179,200),(180,185),(180,196),(180,197),(181,202),(182,202),(183,202),(184,202),(185,202),(186,202),(187,202),(188,202),(189,202),(190,202),(191,202),(192,202),(193,202),(194,202),(195,202),(196,202),(197,202),(198,202),(199,202),(200,202),(201,202)],203)
=> ? = 5 - 1
[1,2,3,4,6,5] => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,16),(1,21),(1,22),(1,28),(1,47),(1,52),(1,53),(1,88),(1,89),(1,91),(2,15),(2,19),(2,20),(2,27),(2,46),(2,50),(2,51),(2,86),(2,87),(2,91),(3,18),(3,24),(3,26),(3,30),(3,49),(3,55),(3,57),(3,87),(3,89),(3,90),(4,17),(4,23),(4,25),(4,29),(4,48),(4,54),(4,56),(4,86),(4,88),(4,90),(5,15),(5,31),(5,32),(5,39),(5,47),(5,54),(5,55),(5,92),(5,93),(5,97),(6,16),(6,33),(6,34),(6,40),(6,46),(6,56),(6,57),(6,94),(6,95),(6,97),(7,17),(7,35),(7,37),(7,41),(7,49),(7,50),(7,52),(7,92),(7,94),(7,96),(8,18),(8,36),(8,38),(8,42),(8,48),(8,51),(8,53),(8,93),(8,95),(8,96),(9,19),(9,23),(9,31),(9,35),(9,44),(9,82),(9,84),(9,89),(9,95),(10,20),(10,24),(10,32),(10,36),(10,45),(10,82),(10,85),(10,88),(10,94),(11,21),(11,25),(11,33),(11,37),(11,45),(11,83),(11,84),(11,87),(11,93),(12,22),(12,26),(12,34),(12,38),(12,44),(12,83),(12,85),(12,86),(12,92),(13,29),(13,30),(13,41),(13,42),(13,43),(13,84),(13,85),(13,91),(13,97),(14,27),(14,28),(14,39),(14,40),(14,43),(14,82),(14,83),(14,90),(14,96),(15,70),(15,71),(15,78),(15,152),(15,153),(15,157),(16,72),(16,73),(16,79),(16,154),(16,155),(16,157),(17,74),(17,76),(17,80),(17,152),(17,154),(17,156),(18,75),(18,77),(18,81),(18,153),(18,155),(18,156),(19,58),(19,70),(19,99),(19,122),(19,134),(19,168),(20,59),(20,71),(20,98),(20,123),(20,134),(20,167),(21,60),(21,72),(21,101),(21,124),(21,135),(21,168),(22,61),(22,73),(22,100),(22,125),(22,135),(22,167),(23,62),(23,74),(23,104),(23,122),(23,136),(23,166),(24,63),(24,75),(24,105),(24,123),(24,137),(24,166),(25,64),(25,76),(25,102),(25,124),(25,136),(25,165),(26,65),(26,77),(26,103),(26,125),(26,137),(26,165),(27,66),(27,78),(27,106),(27,126),(27,134),(27,165),(28,67),(28,79),(28,107),(28,126),(28,135),(28,166),(29,68),(29,80),(29,108),(29,127),(29,136),(29,167),(30,69),(30,81),(30,109),(30,127),(30,137),(30,168),(31,62),(31,70),(31,113),(31,128),(31,138),(31,172),(32,63),(32,71),(32,112),(32,129),(32,138),(32,171),(33,64),(33,72),(33,111),(33,130),(33,139),(33,172),(34,65),(34,73),(34,110),(34,131),(34,139),(34,171),(35,58),(35,74),(35,117),(35,128),(35,140),(35,170),(36,59),(36,75),(36,116),(36,129),(36,141),(36,170),(37,60),(37,76),(37,115),(37,130),(37,140),(37,169),(38,61),(38,77),(38,114),(38,131),(38,141),(38,169),(39,67),(39,78),(39,118),(39,132),(39,138),(39,169),(40,66),(40,79),(40,119),(40,132),(40,139),(40,170),(41,69),(41,80),(41,120),(41,133),(41,140),(41,171),(42,68),(42,81),(42,121),(42,133),(42,141),(42,172),(43,126),(43,127),(43,132),(43,133),(43,158),(44,122),(44,125),(44,128),(44,131),(44,158),(45,123),(45,124),(45,129),(45,130),(45,158),(46,66),(46,98),(46,99),(46,110),(46,111),(46,157),(47,67),(47,100),(47,101),(47,112),(47,113),(47,157),(48,68),(48,102),(48,104),(48,114),(48,116),(48,156),(49,69),(49,103),(49,105),(49,115),(49,117),(49,156),(50,58),(50,98),(50,106),(50,115),(50,120),(50,152),(51,59),(51,99),(51,106),(51,114),(51,121),(51,153),(52,60),(52,100),(52,107),(52,117),(52,120),(52,154),(53,61),(53,101),(53,107),(53,116),(53,121),(53,155),(54,62),(54,102),(54,108),(54,112),(54,118),(54,152),(55,63),(55,103),(55,109),(55,113),(55,118),(55,153),(56,64),(56,104),(56,108),(56,110),(56,119),(56,154),(57,65),(57,105),(57,109),(57,111),(57,119),(57,155),(58,159),(58,173),(58,180),(59,160),(59,173),(59,179),(60,161),(60,174),(60,180),(61,162),(61,174),(61,179),(62,159),(62,175),(62,178),(63,160),(63,176),(63,178),(64,161),(64,175),(64,177),(65,162),(65,176),(65,177),(66,163),(66,173),(66,177),(67,163),(67,174),(67,178),(68,164),(68,175),(68,179),(69,164),(69,176),(69,180),(70,142),(70,159),(70,184),(71,142),(71,160),(71,183),(72,143),(72,161),(72,184),(73,143),(73,162),(73,183),(74,144),(74,159),(74,182),(75,145),(75,160),(75,182),(76,144),(76,161),(76,181),(77,145),(77,162),(77,181),(78,142),(78,163),(78,181),(79,143),(79,163),(79,182),(80,144),(80,164),(80,183),(81,145),(81,164),(81,184),(82,134),(82,138),(82,158),(82,166),(82,170),(83,135),(83,139),(83,158),(83,165),(83,169),(84,136),(84,140),(84,158),(84,168),(84,172),(85,137),(85,141),(85,158),(85,167),(85,171),(86,110),(86,114),(86,122),(86,152),(86,165),(86,167),(87,111),(87,115),(87,123),(87,153),(87,165),(87,168),(88,112),(88,116),(88,124),(88,154),(88,166),(88,167),(89,113),(89,117),(89,125),(89,155),(89,166),(89,168),(90,118),(90,119),(90,127),(90,156),(90,165),(90,166),(91,120),(91,121),(91,126),(91,157),(91,167),(91,168),(92,100),(92,103),(92,128),(92,152),(92,169),(92,171),(93,101),(93,102),(93,129),(93,153),(93,169),(93,172),(94,98),(94,105),(94,130),(94,154),(94,170),(94,171),(95,99),(95,104),(95,131),(95,155),(95,170),(95,172),(96,106),(96,107),(96,133),(96,156),(96,169),(96,170),(97,108),(97,109),(97,132),(97,157),(97,171),(97,172),(98,147),(98,173),(98,183),(99,146),(99,173),(99,184),(100,149),(100,174),(100,183),(101,148),(101,174),(101,184),(102,148),(102,175),(102,181),(103,149),(103,176),(103,181),(104,146),(104,175),(104,182),(105,147),(105,176),(105,182),(106,150),(106,173),(106,181),(107,150),(107,174),(107,182),(108,151),(108,175),(108,183),(109,151),(109,176),(109,184),(110,146),(110,177),(110,183),(111,147),(111,177),(111,184),(112,148),(112,178),(112,183),(113,149),(113,178),(113,184),(114,146),(114,179),(114,181),(115,147),(115,180),(115,181),(116,148),(116,179),(116,182),(117,149),(117,180),(117,182),(118,151),(118,178),(118,181),(119,151),(119,177),(119,182),(120,150),(120,180),(120,183),(121,150),(121,179),(121,184),(122,146),(122,159),(122,186),(123,147),(123,160),(123,186),(124,148),(124,161),(124,186),(125,149),(125,162),(125,186),(126,150),(126,163),(126,186),(127,151),(127,164),(127,186),(128,149),(128,159),(128,185),(129,148),(129,160),(129,185),(130,147),(130,161),(130,185),(131,146),(131,162),(131,185),(132,151),(132,163),(132,185),(133,150),(133,164),(133,185),(134,142),(134,173),(134,186),(135,143),(135,174),(135,186),(136,144),(136,175),(136,186),(137,145),(137,176),(137,186),(138,142),(138,178),(138,185),(139,143),(139,177),(139,185),(140,144),(140,180),(140,185),(141,145),(141,179),(141,185),(142,187),(143,187),(144,187),(145,187),(146,187),(147,187),(148,187),(149,187),(150,187),(151,187),(152,159),(152,181),(152,183),(153,160),(153,181),(153,184),(154,161),(154,182),(154,183),(155,162),(155,182),(155,184),(156,164),(156,181),(156,182),(157,163),(157,183),(157,184),(158,185),(158,186),(159,187),(160,187),(161,187),(162,187),(163,187),(164,187),(165,177),(165,181),(165,186),(166,178),(166,182),(166,186),(167,179),(167,183),(167,186),(168,180),(168,184),(168,186),(169,174),(169,181),(169,185),(170,173),(170,182),(170,185),(171,176),(171,183),(171,185),(172,175),(172,184),(172,185),(173,187),(174,187),(175,187),(176,187),(177,187),(178,187),(179,187),(180,187),(181,187),(182,187),(183,187),(184,187),(185,187),(186,187)],188)
=> ? = 5 - 1
[1,2,3,6,4,5] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,25),(1,30),(1,31),(1,36),(1,37),(1,40),(1,43),(1,46),(1,64),(1,65),(2,24),(2,27),(2,29),(2,33),(2,35),(2,39),(2,42),(2,45),(2,63),(2,65),(3,23),(3,26),(3,28),(3,32),(3,34),(3,38),(3,41),(3,44),(3,63),(3,64),(4,20),(4,21),(4,22),(4,23),(4,24),(4,25),(4,90),(4,91),(4,92),(5,16),(5,28),(5,29),(5,43),(5,47),(5,48),(5,84),(5,85),(5,90),(6,14),(6,26),(6,30),(6,42),(6,49),(6,51),(6,84),(6,86),(6,91),(7,15),(7,27),(7,31),(7,41),(7,50),(7,52),(7,85),(7,86),(7,92),(8,19),(8,34),(8,35),(8,46),(8,49),(8,50),(8,87),(8,88),(8,90),(9,17),(9,32),(9,36),(9,45),(9,47),(9,52),(9,87),(9,89),(9,91),(10,18),(10,33),(10,37),(10,44),(10,48),(10,51),(10,88),(10,89),(10,92),(11,15),(11,18),(11,20),(11,38),(11,62),(11,65),(11,84),(11,87),(12,14),(12,17),(12,21),(12,39),(12,62),(12,64),(12,85),(12,88),(13,16),(13,19),(13,22),(13,40),(13,62),(13,63),(13,86),(13,89),(14,67),(14,94),(14,97),(14,112),(14,146),(15,66),(15,93),(15,98),(15,113),(15,146),(16,68),(16,95),(16,96),(16,114),(16,146),(17,70),(17,94),(17,100),(17,116),(17,145),(18,69),(18,93),(18,101),(18,115),(18,145),(19,71),(19,95),(19,99),(19,117),(19,145),(20,59),(20,93),(20,102),(20,111),(20,148),(21,60),(21,94),(21,102),(21,110),(21,149),(22,61),(22,95),(22,102),(22,109),(22,150),(23,59),(23,103),(23,105),(23,109),(23,110),(23,124),(24,60),(24,104),(24,106),(24,109),(24,111),(24,125),(25,61),(25,107),(25,108),(25,110),(25,111),(25,126),(26,53),(26,80),(26,97),(26,103),(26,118),(26,139),(27,54),(27,81),(27,98),(27,104),(27,119),(27,139),(28,55),(28,78),(28,96),(28,105),(28,118),(28,140),(29,56),(29,79),(29,96),(29,106),(29,119),(29,141),(30,57),(30,83),(30,97),(30,107),(30,120),(30,141),(31,58),(31,82),(31,98),(31,108),(31,120),(31,140),(32,55),(32,74),(32,100),(32,103),(32,121),(32,142),(33,56),(33,75),(33,101),(33,104),(33,122),(33,142),(34,53),(34,72),(34,99),(34,105),(34,121),(34,143),(35,54),(35,73),(35,99),(35,106),(35,122),(35,144),(36,58),(36,77),(36,100),(36,107),(36,123),(36,144),(37,57),(37,76),(37,101),(37,108),(37,123),(37,143),(38,59),(38,66),(38,69),(38,118),(38,121),(38,147),(39,60),(39,67),(39,70),(39,119),(39,122),(39,147),(40,61),(40,68),(40,71),(40,120),(40,123),(40,147),(41,66),(41,72),(41,74),(41,124),(41,139),(41,140),(42,67),(42,73),(42,75),(42,125),(42,139),(42,141),(43,68),(43,76),(43,77),(43,126),(43,140),(43,141),(44,69),(44,78),(44,80),(44,124),(44,142),(44,143),(45,70),(45,79),(45,81),(45,125),(45,142),(45,144),(46,71),(46,82),(46,83),(46,126),(46,143),(46,144),(47,55),(47,77),(47,79),(47,114),(47,116),(47,148),(48,56),(48,76),(48,78),(48,114),(48,115),(48,149),(49,53),(49,73),(49,83),(49,112),(49,117),(49,148),(50,54),(50,72),(50,82),(50,113),(50,117),(50,149),(51,57),(51,75),(51,80),(51,112),(51,115),(51,150),(52,58),(52,74),(52,81),(52,113),(52,116),(52,150),(53,152),(53,154),(53,158),(54,153),(54,154),(54,159),(55,151),(55,155),(55,158),(56,151),(56,156),(56,159),(57,152),(57,156),(57,160),(58,153),(58,155),(58,160),(59,127),(59,157),(59,158),(60,128),(60,157),(60,159),(61,129),(61,157),(61,160),(62,102),(62,145),(62,146),(62,147),(63,96),(63,99),(63,109),(63,139),(63,142),(63,147),(64,97),(64,100),(64,110),(64,140),(64,143),(64,147),(65,98),(65,101),(65,111),(65,141),(65,144),(65,147),(66,127),(66,130),(66,164),(67,128),(67,131),(67,164),(68,129),(68,132),(68,164),(69,127),(69,133),(69,165),(70,128),(70,134),(70,165),(71,129),(71,135),(71,165),(72,130),(72,154),(72,162),(73,131),(73,154),(73,163),(74,130),(74,155),(74,161),(75,131),(75,156),(75,161),(76,132),(76,156),(76,162),(77,132),(77,155),(77,163),(78,133),(78,151),(78,162),(79,134),(79,151),(79,163),(80,133),(80,152),(80,161),(81,134),(81,153),(81,161),(82,135),(82,153),(82,162),(83,135),(83,152),(83,163),(84,115),(84,118),(84,141),(84,146),(84,148),(85,116),(85,119),(85,140),(85,146),(85,149),(86,117),(86,120),(86,139),(86,146),(86,150),(87,113),(87,121),(87,144),(87,145),(87,148),(88,112),(88,122),(88,143),(88,145),(88,149),(89,114),(89,123),(89,142),(89,145),(89,150),(90,95),(90,105),(90,106),(90,126),(90,148),(90,149),(91,94),(91,103),(91,107),(91,125),(91,148),(91,150),(92,93),(92,104),(92,108),(92,124),(92,149),(92,150),(93,127),(93,138),(93,166),(94,128),(94,137),(94,166),(95,129),(95,136),(95,166),(96,136),(96,151),(96,164),(97,137),(97,152),(97,164),(98,138),(98,153),(98,164),(99,136),(99,154),(99,165),(100,137),(100,155),(100,165),(101,138),(101,156),(101,165),(102,157),(102,166),(103,137),(103,158),(103,161),(104,138),(104,159),(104,161),(105,136),(105,158),(105,162),(106,136),(106,159),(106,163),(107,137),(107,160),(107,163),(108,138),(108,160),(108,162),(109,136),(109,157),(109,161),(110,137),(110,157),(110,162),(111,138),(111,157),(111,163),(112,131),(112,152),(112,166),(113,130),(113,153),(113,166),(114,132),(114,151),(114,166),(115,133),(115,156),(115,166),(116,134),(116,155),(116,166),(117,135),(117,154),(117,166),(118,133),(118,158),(118,164),(119,134),(119,159),(119,164),(120,135),(120,160),(120,164),(121,130),(121,158),(121,165),(122,131),(122,159),(122,165),(123,132),(123,160),(123,165),(124,127),(124,161),(124,162),(125,128),(125,161),(125,163),(126,129),(126,162),(126,163),(127,167),(128,167),(129,167),(130,167),(131,167),(132,167),(133,167),(134,167),(135,167),(136,167),(137,167),(138,167),(139,154),(139,161),(139,164),(140,155),(140,162),(140,164),(141,156),(141,163),(141,164),(142,151),(142,161),(142,165),(143,152),(143,162),(143,165),(144,153),(144,163),(144,165),(145,165),(145,166),(146,164),(146,166),(147,157),(147,164),(147,165),(148,158),(148,163),(148,166),(149,159),(149,162),(149,166),(150,160),(150,161),(150,166),(151,167),(152,167),(153,167),(154,167),(155,167),(156,167),(157,167),(158,167),(159,167),(160,167),(161,167),(162,167),(163,167),(164,167),(165,167),(166,167)],168)
=> ? = 5 - 1
[1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,19),(1,20),(1,21),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(1,46),(2,16),(2,17),(2,18),(2,34),(2,35),(2,36),(2,37),(2,38),(2,39),(2,46),(3,15),(3,18),(3,21),(3,30),(3,33),(3,76),(3,79),(3,80),(4,14),(4,17),(4,20),(4,29),(4,32),(4,75),(4,78),(4,80),(5,13),(5,16),(5,19),(5,28),(5,31),(5,74),(5,77),(5,80),(6,23),(6,26),(6,28),(6,34),(6,40),(6,71),(6,75),(6,76),(7,22),(7,27),(7,29),(7,35),(7,41),(7,72),(7,74),(7,76),(8,24),(8,25),(8,30),(8,36),(8,42),(8,73),(8,74),(8,75),(9,25),(9,27),(9,31),(9,37),(9,43),(9,71),(9,78),(9,79),(10,24),(10,26),(10,32),(10,38),(10,44),(10,72),(10,77),(10,79),(11,22),(11,23),(11,33),(11,39),(11,45),(11,73),(11,77),(11,78),(12,13),(12,14),(12,15),(12,46),(12,71),(12,72),(12,73),(13,81),(13,84),(13,99),(13,127),(14,81),(14,85),(14,100),(14,128),(15,81),(15,86),(15,101),(15,129),(16,47),(16,50),(16,82),(16,84),(16,87),(16,90),(17,48),(17,51),(17,82),(17,85),(17,88),(17,91),(18,49),(18,52),(18,82),(18,86),(18,89),(18,92),(19,53),(19,56),(19,83),(19,84),(19,93),(19,96),(20,54),(20,57),(20,83),(20,85),(20,94),(20,97),(21,55),(21,58),(21,83),(21,86),(21,95),(21,98),(22,59),(22,65),(22,103),(22,107),(22,127),(23,60),(23,66),(23,102),(23,107),(23,128),(24,61),(24,67),(24,104),(24,106),(24,127),(25,62),(25,68),(25,104),(25,105),(25,128),(26,63),(26,69),(26,102),(26,106),(26,129),(27,64),(27,70),(27,103),(27,105),(27,129),(28,47),(28,53),(28,99),(28,102),(28,131),(29,48),(29,54),(29,100),(29,103),(29,131),(30,49),(30,55),(30,101),(30,104),(30,131),(31,50),(31,56),(31,99),(31,105),(31,130),(32,51),(32,57),(32,100),(32,106),(32,130),(33,52),(33,58),(33,101),(33,107),(33,130),(34,47),(34,60),(34,63),(34,88),(34,89),(34,120),(35,48),(35,59),(35,64),(35,87),(35,89),(35,121),(36,49),(36,61),(36,62),(36,87),(36,88),(36,122),(37,50),(37,62),(37,64),(37,91),(37,92),(37,120),(38,51),(38,61),(38,63),(38,90),(38,92),(38,121),(39,52),(39,59),(39,60),(39,90),(39,91),(39,122),(40,53),(40,66),(40,69),(40,94),(40,95),(40,120),(41,54),(41,65),(41,70),(41,93),(41,95),(41,121),(42,55),(42,67),(42,68),(42,93),(42,94),(42,122),(43,56),(43,68),(43,70),(43,97),(43,98),(43,120),(44,57),(44,67),(44,69),(44,96),(44,98),(44,121),(45,58),(45,65),(45,66),(45,96),(45,97),(45,122),(46,84),(46,85),(46,86),(46,120),(46,121),(46,122),(47,108),(47,124),(47,132),(48,109),(48,125),(48,132),(49,110),(49,126),(49,132),(50,111),(50,124),(50,133),(51,112),(51,125),(51,133),(52,113),(52,126),(52,133),(53,114),(53,124),(53,134),(54,115),(54,125),(54,134),(55,116),(55,126),(55,134),(56,117),(56,124),(56,135),(57,118),(57,125),(57,135),(58,119),(58,126),(58,135),(59,109),(59,113),(59,136),(60,108),(60,113),(60,137),(61,110),(61,112),(61,136),(62,110),(62,111),(62,137),(63,108),(63,112),(63,138),(64,109),(64,111),(64,138),(65,115),(65,119),(65,136),(66,114),(66,119),(66,137),(67,116),(67,118),(67,136),(68,116),(68,117),(68,137),(69,114),(69,118),(69,138),(70,115),(70,117),(70,138),(71,99),(71,120),(71,128),(71,129),(72,100),(72,121),(72,127),(72,129),(73,101),(73,122),(73,127),(73,128),(74,87),(74,93),(74,105),(74,127),(74,131),(75,88),(75,94),(75,106),(75,128),(75,131),(76,89),(76,95),(76,107),(76,129),(76,131),(77,90),(77,96),(77,102),(77,127),(77,130),(78,91),(78,97),(78,103),(78,128),(78,130),(79,92),(79,98),(79,104),(79,129),(79,130),(80,81),(80,82),(80,83),(80,130),(80,131),(81,123),(81,139),(82,123),(82,132),(82,133),(83,123),(83,134),(83,135),(84,123),(84,124),(84,136),(85,123),(85,125),(85,137),(86,123),(86,126),(86,138),(87,111),(87,132),(87,136),(88,112),(88,132),(88,137),(89,113),(89,132),(89,138),(90,108),(90,133),(90,136),(91,109),(91,133),(91,137),(92,110),(92,133),(92,138),(93,117),(93,134),(93,136),(94,118),(94,134),(94,137),(95,119),(95,134),(95,138),(96,114),(96,135),(96,136),(97,115),(97,135),(97,137),(98,116),(98,135),(98,138),(99,124),(99,139),(100,125),(100,139),(101,126),(101,139),(102,108),(102,114),(102,139),(103,109),(103,115),(103,139),(104,110),(104,116),(104,139),(105,111),(105,117),(105,139),(106,112),(106,118),(106,139),(107,113),(107,119),(107,139),(108,140),(109,140),(110,140),(111,140),(112,140),(113,140),(114,140),(115,140),(116,140),(117,140),(118,140),(119,140),(120,124),(120,137),(120,138),(121,125),(121,136),(121,138),(122,126),(122,136),(122,137),(123,140),(124,140),(125,140),(126,140),(127,136),(127,139),(128,137),(128,139),(129,138),(129,139),(130,133),(130,135),(130,139),(131,132),(131,134),(131,139),(132,140),(133,140),(134,140),(135,140),(136,140),(137,140),(138,140),(139,140)],141)
=> ? = 5 - 1
[1,3,4,5,2,6] => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,19),(1,20),(1,21),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(1,46),(2,16),(2,17),(2,18),(2,34),(2,35),(2,36),(2,37),(2,38),(2,39),(2,46),(3,15),(3,18),(3,21),(3,30),(3,33),(3,76),(3,79),(3,80),(4,14),(4,17),(4,20),(4,29),(4,32),(4,75),(4,78),(4,80),(5,13),(5,16),(5,19),(5,28),(5,31),(5,74),(5,77),(5,80),(6,23),(6,26),(6,28),(6,34),(6,40),(6,71),(6,75),(6,76),(7,22),(7,27),(7,29),(7,35),(7,41),(7,72),(7,74),(7,76),(8,24),(8,25),(8,30),(8,36),(8,42),(8,73),(8,74),(8,75),(9,25),(9,27),(9,31),(9,37),(9,43),(9,71),(9,78),(9,79),(10,24),(10,26),(10,32),(10,38),(10,44),(10,72),(10,77),(10,79),(11,22),(11,23),(11,33),(11,39),(11,45),(11,73),(11,77),(11,78),(12,13),(12,14),(12,15),(12,46),(12,71),(12,72),(12,73),(13,81),(13,84),(13,99),(13,127),(14,81),(14,85),(14,100),(14,128),(15,81),(15,86),(15,101),(15,129),(16,47),(16,50),(16,82),(16,84),(16,87),(16,90),(17,48),(17,51),(17,82),(17,85),(17,88),(17,91),(18,49),(18,52),(18,82),(18,86),(18,89),(18,92),(19,53),(19,56),(19,83),(19,84),(19,93),(19,96),(20,54),(20,57),(20,83),(20,85),(20,94),(20,97),(21,55),(21,58),(21,83),(21,86),(21,95),(21,98),(22,59),(22,65),(22,103),(22,107),(22,127),(23,60),(23,66),(23,102),(23,107),(23,128),(24,61),(24,67),(24,104),(24,106),(24,127),(25,62),(25,68),(25,104),(25,105),(25,128),(26,63),(26,69),(26,102),(26,106),(26,129),(27,64),(27,70),(27,103),(27,105),(27,129),(28,47),(28,53),(28,99),(28,102),(28,131),(29,48),(29,54),(29,100),(29,103),(29,131),(30,49),(30,55),(30,101),(30,104),(30,131),(31,50),(31,56),(31,99),(31,105),(31,130),(32,51),(32,57),(32,100),(32,106),(32,130),(33,52),(33,58),(33,101),(33,107),(33,130),(34,47),(34,60),(34,63),(34,88),(34,89),(34,120),(35,48),(35,59),(35,64),(35,87),(35,89),(35,121),(36,49),(36,61),(36,62),(36,87),(36,88),(36,122),(37,50),(37,62),(37,64),(37,91),(37,92),(37,120),(38,51),(38,61),(38,63),(38,90),(38,92),(38,121),(39,52),(39,59),(39,60),(39,90),(39,91),(39,122),(40,53),(40,66),(40,69),(40,94),(40,95),(40,120),(41,54),(41,65),(41,70),(41,93),(41,95),(41,121),(42,55),(42,67),(42,68),(42,93),(42,94),(42,122),(43,56),(43,68),(43,70),(43,97),(43,98),(43,120),(44,57),(44,67),(44,69),(44,96),(44,98),(44,121),(45,58),(45,65),(45,66),(45,96),(45,97),(45,122),(46,84),(46,85),(46,86),(46,120),(46,121),(46,122),(47,108),(47,124),(47,132),(48,109),(48,125),(48,132),(49,110),(49,126),(49,132),(50,111),(50,124),(50,133),(51,112),(51,125),(51,133),(52,113),(52,126),(52,133),(53,114),(53,124),(53,134),(54,115),(54,125),(54,134),(55,116),(55,126),(55,134),(56,117),(56,124),(56,135),(57,118),(57,125),(57,135),(58,119),(58,126),(58,135),(59,109),(59,113),(59,136),(60,108),(60,113),(60,137),(61,110),(61,112),(61,136),(62,110),(62,111),(62,137),(63,108),(63,112),(63,138),(64,109),(64,111),(64,138),(65,115),(65,119),(65,136),(66,114),(66,119),(66,137),(67,116),(67,118),(67,136),(68,116),(68,117),(68,137),(69,114),(69,118),(69,138),(70,115),(70,117),(70,138),(71,99),(71,120),(71,128),(71,129),(72,100),(72,121),(72,127),(72,129),(73,101),(73,122),(73,127),(73,128),(74,87),(74,93),(74,105),(74,127),(74,131),(75,88),(75,94),(75,106),(75,128),(75,131),(76,89),(76,95),(76,107),(76,129),(76,131),(77,90),(77,96),(77,102),(77,127),(77,130),(78,91),(78,97),(78,103),(78,128),(78,130),(79,92),(79,98),(79,104),(79,129),(79,130),(80,81),(80,82),(80,83),(80,130),(80,131),(81,123),(81,139),(82,123),(82,132),(82,133),(83,123),(83,134),(83,135),(84,123),(84,124),(84,136),(85,123),(85,125),(85,137),(86,123),(86,126),(86,138),(87,111),(87,132),(87,136),(88,112),(88,132),(88,137),(89,113),(89,132),(89,138),(90,108),(90,133),(90,136),(91,109),(91,133),(91,137),(92,110),(92,133),(92,138),(93,117),(93,134),(93,136),(94,118),(94,134),(94,137),(95,119),(95,134),(95,138),(96,114),(96,135),(96,136),(97,115),(97,135),(97,137),(98,116),(98,135),(98,138),(99,124),(99,139),(100,125),(100,139),(101,126),(101,139),(102,108),(102,114),(102,139),(103,109),(103,115),(103,139),(104,110),(104,116),(104,139),(105,111),(105,117),(105,139),(106,112),(106,118),(106,139),(107,113),(107,119),(107,139),(108,140),(109,140),(110,140),(111,140),(112,140),(113,140),(114,140),(115,140),(116,140),(117,140),(118,140),(119,140),(120,124),(120,137),(120,138),(121,125),(121,136),(121,138),(122,126),(122,136),(122,137),(123,140),(124,140),(125,140),(126,140),(127,136),(127,139),(128,137),(128,139),(129,138),(129,139),(130,133),(130,135),(130,139),(131,132),(131,134),(131,139),(132,140),(133,140),(134,140),(135,140),(136,140),(137,140),(138,140),(139,140)],141)
=> ? = 3 - 1
[1,3,4,5,6,2] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,13),(1,24),(1,25),(1,31),(1,54),(1,55),(1,57),(2,12),(2,22),(2,23),(2,30),(2,52),(2,53),(2,57),(3,15),(3,27),(3,29),(3,33),(3,53),(3,55),(3,56),(4,14),(4,26),(4,28),(4,32),(4,52),(4,54),(4,56),(5,16),(5,22),(5,26),(5,35),(5,55),(5,58),(5,60),(6,17),(6,23),(6,27),(6,36),(6,54),(6,58),(6,61),(7,18),(7,24),(7,28),(7,36),(7,53),(7,59),(7,60),(8,19),(8,25),(8,29),(8,35),(8,52),(8,59),(8,61),(9,21),(9,32),(9,33),(9,34),(9,57),(9,60),(9,61),(10,20),(10,30),(10,31),(10,34),(10,56),(10,58),(10,59),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(12,37),(12,38),(12,45),(12,72),(12,73),(12,77),(13,39),(13,40),(13,46),(13,74),(13,75),(13,77),(14,41),(14,43),(14,47),(14,72),(14,74),(14,76),(15,42),(15,44),(15,48),(15,73),(15,75),(15,76),(16,37),(16,41),(16,49),(16,75),(16,78),(16,80),(17,38),(17,42),(17,50),(17,74),(17,78),(17,81),(18,39),(18,43),(18,50),(18,73),(18,79),(18,80),(19,40),(19,44),(19,49),(19,72),(19,79),(19,81),(20,45),(20,46),(20,51),(20,76),(20,78),(20,79),(21,47),(21,48),(21,51),(21,77),(21,80),(21,81),(22,37),(22,62),(22,66),(22,96),(23,38),(23,62),(23,67),(23,95),(24,39),(24,63),(24,68),(24,96),(25,40),(25,63),(25,69),(25,95),(26,41),(26,64),(26,66),(26,94),(27,42),(27,65),(27,67),(27,94),(28,43),(28,64),(28,68),(28,93),(29,44),(29,65),(29,69),(29,93),(30,45),(30,62),(30,70),(30,93),(31,46),(31,63),(31,70),(31,94),(32,47),(32,64),(32,71),(32,95),(33,48),(33,65),(33,71),(33,96),(34,51),(34,70),(34,71),(34,92),(35,49),(35,66),(35,69),(35,92),(36,50),(36,67),(36,68),(36,92),(37,82),(37,86),(37,100),(38,82),(38,87),(38,99),(39,83),(39,88),(39,100),(40,83),(40,89),(40,99),(41,84),(41,86),(41,98),(42,85),(42,87),(42,98),(43,84),(43,88),(43,97),(44,85),(44,89),(44,97),(45,82),(45,90),(45,97),(46,83),(46,90),(46,98),(47,84),(47,91),(47,99),(48,85),(48,91),(48,100),(49,86),(49,89),(49,101),(50,87),(50,88),(50,101),(51,90),(51,91),(51,101),(52,66),(52,72),(52,93),(52,95),(53,67),(53,73),(53,93),(53,96),(54,68),(54,74),(54,94),(54,95),(55,69),(55,75),(55,94),(55,96),(56,71),(56,76),(56,93),(56,94),(57,70),(57,77),(57,95),(57,96),(58,62),(58,78),(58,92),(58,94),(59,63),(59,79),(59,92),(59,93),(60,64),(60,80),(60,92),(60,96),(61,65),(61,81),(61,92),(61,95),(62,82),(62,102),(63,83),(63,102),(64,84),(64,102),(65,85),(65,102),(66,86),(66,102),(67,87),(67,102),(68,88),(68,102),(69,89),(69,102),(70,90),(70,102),(71,91),(71,102),(72,86),(72,97),(72,99),(73,87),(73,97),(73,100),(74,88),(74,98),(74,99),(75,89),(75,98),(75,100),(76,91),(76,97),(76,98),(77,90),(77,99),(77,100),(78,82),(78,98),(78,101),(79,83),(79,97),(79,101),(80,84),(80,100),(80,101),(81,85),(81,99),(81,101),(82,103),(83,103),(84,103),(85,103),(86,103),(87,103),(88,103),(89,103),(90,103),(91,103),(92,101),(92,102),(93,97),(93,102),(94,98),(94,102),(95,99),(95,102),(96,100),(96,102),(97,103),(98,103),(99,103),(100,103),(101,103),(102,103)],104)
=> ? = 3 - 1
[1,3,4,6,5,2] => [2,5,6,4,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(1,18),(1,19),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,15),(2,16),(2,22),(2,28),(2,31),(2,55),(2,56),(3,12),(3,14),(3,21),(3,27),(3,30),(3,54),(3,56),(4,11),(4,13),(4,20),(4,26),(4,29),(4,54),(4,55),(5,13),(5,14),(5,25),(5,28),(5,32),(5,57),(5,58),(6,11),(6,15),(6,23),(6,27),(6,33),(6,57),(6,59),(7,12),(7,16),(7,24),(7,26),(7,34),(7,58),(7,59),(8,17),(8,20),(8,24),(8,41),(8,56),(8,57),(9,18),(9,21),(9,23),(9,41),(9,55),(9,58),(10,19),(10,22),(10,25),(10,41),(10,54),(10,59),(11,35),(11,70),(11,74),(11,82),(12,36),(12,71),(12,75),(12,82),(13,37),(13,70),(13,73),(13,83),(14,38),(14,71),(14,73),(14,84),(15,39),(15,72),(15,74),(15,84),(16,40),(16,72),(16,75),(16,83),(17,45),(17,49),(17,63),(17,66),(17,67),(18,46),(18,48),(18,63),(18,65),(18,68),(19,47),(19,50),(19,63),(19,64),(19,69),(20,45),(20,51),(20,70),(20,89),(21,46),(21,52),(21,71),(21,89),(22,47),(22,53),(22,72),(22,89),(23,48),(23,52),(23,74),(23,88),(24,49),(24,51),(24,75),(24,88),(25,50),(25,53),(25,73),(25,88),(26,42),(26,51),(26,82),(26,83),(27,43),(27,52),(27,82),(27,84),(28,44),(28,53),(28,83),(28,84),(29,35),(29,37),(29,42),(29,45),(29,64),(29,65),(30,36),(30,38),(30,43),(30,46),(30,64),(30,66),(31,39),(31,40),(31,44),(31,47),(31,65),(31,66),(32,37),(32,38),(32,44),(32,50),(32,67),(32,68),(33,35),(33,39),(33,43),(33,48),(33,67),(33,69),(34,36),(34,40),(34,42),(34,49),(34,68),(34,69),(35,76),(35,80),(35,85),(36,77),(36,81),(36,85),(37,76),(37,79),(37,86),(38,77),(38,79),(38,87),(39,78),(39,80),(39,87),(40,78),(40,81),(40,86),(41,63),(41,88),(41,89),(42,60),(42,85),(42,86),(43,61),(43,85),(43,87),(44,62),(44,86),(44,87),(45,60),(45,76),(45,90),(46,61),(46,77),(46,90),(47,62),(47,78),(47,90),(48,61),(48,80),(48,91),(49,60),(49,81),(49,91),(50,62),(50,79),(50,91),(51,60),(51,92),(52,61),(52,92),(53,62),(53,92),(54,64),(54,73),(54,82),(54,89),(55,65),(55,74),(55,83),(55,89),(56,66),(56,75),(56,84),(56,89),(57,67),(57,70),(57,84),(57,88),(58,68),(58,71),(58,83),(58,88),(59,69),(59,72),(59,82),(59,88),(60,93),(61,93),(62,93),(63,90),(63,91),(64,79),(64,85),(64,90),(65,80),(65,86),(65,90),(66,81),(66,87),(66,90),(67,76),(67,87),(67,91),(68,77),(68,86),(68,91),(69,78),(69,85),(69,91),(70,76),(70,92),(71,77),(71,92),(72,78),(72,92),(73,79),(73,92),(74,80),(74,92),(75,81),(75,92),(76,93),(77,93),(78,93),(79,93),(80,93),(81,93),(82,85),(82,92),(83,86),(83,92),(84,87),(84,92),(85,93),(86,93),(87,93),(88,91),(88,92),(89,90),(89,92),(90,93),(91,93),(92,93)],94)
=> ? = 3 - 1
[1,3,5,2,4,6] => [6,4,2,5,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,20),(1,29),(1,31),(1,36),(1,40),(1,42),(1,44),(1,81),(1,86),(2,19),(2,28),(2,30),(2,35),(2,39),(2,41),(2,43),(2,81),(2,85),(3,17),(3,26),(3,32),(3,35),(3,37),(3,42),(3,45),(3,82),(3,84),(4,18),(4,27),(4,33),(4,36),(4,38),(4,41),(4,46),(4,82),(4,83),(5,25),(5,37),(5,38),(5,39),(5,40),(5,47),(5,48),(5,79),(5,80),(6,15),(6,16),(6,24),(6,32),(6,33),(6,34),(6,48),(6,85),(6,86),(7,13),(7,14),(7,23),(7,30),(7,31),(7,34),(7,47),(7,83),(7,84),(8,13),(8,17),(8,22),(8,27),(8,43),(8,49),(8,79),(8,86),(9,14),(9,18),(9,21),(9,26),(9,44),(9,49),(9,80),(9,85),(10,15),(10,19),(10,21),(10,29),(10,45),(10,50),(10,79),(10,83),(11,16),(11,20),(11,22),(11,28),(11,46),(11,50),(11,80),(11,84),(12,23),(12,24),(12,25),(12,49),(12,50),(12,81),(12,82),(13,63),(13,91),(13,103),(13,108),(13,127),(14,64),(14,90),(14,102),(14,108),(14,128),(15,65),(15,92),(15,104),(15,109),(15,130),(16,66),(16,93),(16,105),(16,109),(16,129),(17,59),(17,96),(17,101),(17,103),(17,131),(18,60),(18,97),(18,100),(18,102),(18,131),(19,61),(19,94),(19,98),(19,104),(19,132),(20,62),(20,95),(20,99),(20,105),(20,132),(21,73),(21,74),(21,102),(21,104),(21,138),(22,72),(22,75),(22,103),(22,105),(22,138),(23,76),(23,78),(23,108),(23,110),(23,139),(24,77),(24,78),(24,109),(24,111),(24,140),(25,76),(25,77),(25,106),(25,107),(25,138),(26,69),(26,74),(26,114),(26,128),(26,131),(27,68),(27,75),(27,115),(27,127),(27,131),(28,70),(28,72),(28,112),(28,129),(28,132),(29,71),(29,73),(29,113),(29,130),(29,132),(30,53),(30,63),(30,90),(30,98),(30,110),(30,112),(31,54),(31,64),(31,91),(31,99),(31,110),(31,113),(32,56),(32,65),(32,93),(32,101),(32,111),(32,114),(33,55),(33,66),(33,92),(33,100),(33,111),(33,115),(34,67),(34,78),(34,90),(34,91),(34,92),(34,93),(35,51),(35,59),(35,61),(35,112),(35,114),(35,124),(36,52),(36,60),(36,62),(36,113),(36,115),(36,124),(37,51),(37,56),(37,58),(37,96),(37,106),(37,128),(38,52),(38,55),(38,57),(38,97),(38,106),(38,127),(39,51),(39,53),(39,57),(39,94),(39,107),(39,129),(40,52),(40,54),(40,58),(40,95),(40,107),(40,130),(41,57),(41,68),(41,70),(41,98),(41,100),(41,124),(42,58),(42,69),(42,71),(42,99),(42,101),(42,124),(43,59),(43,63),(43,68),(43,72),(43,94),(43,140),(44,60),(44,64),(44,69),(44,73),(44,95),(44,140),(45,61),(45,65),(45,71),(45,74),(45,96),(45,139),(46,62),(46,66),(46,70),(46,75),(46,97),(46,139),(47,53),(47,54),(47,67),(47,76),(47,127),(47,128),(48,55),(48,56),(48,67),(48,77),(48,129),(48,130),(49,108),(49,131),(49,138),(49,140),(50,109),(50,132),(50,138),(50,139),(51,122),(51,133),(51,143),(52,123),(52,133),(52,144),(53,116),(53,134),(53,143),(54,116),(54,135),(54,144),(55,117),(55,136),(55,144),(56,117),(56,137),(56,143),(57,133),(57,134),(57,136),(58,133),(58,135),(58,137),(59,118),(59,122),(59,146),(60,119),(60,123),(60,146),(61,120),(61,122),(61,145),(62,121),(62,123),(62,145),(63,118),(63,134),(63,141),(64,119),(64,135),(64,141),(65,120),(65,137),(65,142),(66,121),(66,136),(66,142),(67,87),(67,143),(67,144),(68,88),(68,134),(68,146),(69,89),(69,135),(69,146),(70,88),(70,136),(70,145),(71,89),(71,137),(71,145),(72,88),(72,118),(72,148),(73,89),(73,119),(73,148),(74,89),(74,120),(74,147),(75,88),(75,121),(75,147),(76,87),(76,116),(76,147),(77,87),(77,117),(77,148),(78,87),(78,141),(78,142),(79,94),(79,96),(79,127),(79,130),(79,138),(80,95),(80,97),(80,128),(80,129),(80,138),(81,107),(81,110),(81,124),(81,132),(81,140),(82,106),(82,111),(82,124),(82,131),(82,139),(83,92),(83,98),(83,102),(83,113),(83,127),(83,139),(84,93),(84,99),(84,103),(84,112),(84,128),(84,139),(85,90),(85,100),(85,104),(85,114),(85,129),(85,140),(86,91),(86,101),(86,105),(86,115),(86,130),(86,140),(87,149),(88,149),(89,149),(90,125),(90,141),(90,143),(91,126),(91,141),(91,144),(92,125),(92,142),(92,144),(93,126),(93,142),(93,143),(94,122),(94,134),(94,148),(95,123),(95,135),(95,148),(96,122),(96,137),(96,147),(97,123),(97,136),(97,147),(98,125),(98,134),(98,145),(99,126),(99,135),(99,145),(100,125),(100,136),(100,146),(101,126),(101,137),(101,146),(102,119),(102,125),(102,147),(103,118),(103,126),(103,147),(104,120),(104,125),(104,148),(105,121),(105,126),(105,148),(106,117),(106,133),(106,147),(107,116),(107,133),(107,148),(108,141),(108,147),(109,142),(109,148),(110,116),(110,141),(110,145),(111,117),(111,142),(111,146),(112,118),(112,143),(112,145),(113,119),(113,144),(113,145),(114,120),(114,143),(114,146),(115,121),(115,144),(115,146),(116,149),(117,149),(118,149),(119,149),(120,149),(121,149),(122,149),(123,149),(124,133),(124,145),(124,146),(125,149),(126,149),(127,134),(127,144),(127,147),(128,135),(128,143),(128,147),(129,136),(129,143),(129,148),(130,137),(130,144),(130,148),(131,146),(131,147),(132,145),(132,148),(133,149),(134,149),(135,149),(136,149),(137,149),(138,147),(138,148),(139,142),(139,145),(139,147),(140,141),(140,146),(140,148),(141,149),(142,149),(143,149),(144,149),(145,149),(146,149),(147,149),(148,149)],150)
=> ? = 3 - 1
[1,3,5,2,6,4] => [4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,18),(1,22),(1,29),(1,36),(1,38),(1,40),(1,42),(1,44),(1,78),(2,17),(2,21),(2,29),(2,35),(2,37),(2,39),(2,41),(2,43),(2,77),(3,14),(3,19),(3,20),(3,32),(3,33),(3,34),(3,37),(3,38),(3,79),(4,15),(4,27),(4,31),(4,33),(4,35),(4,40),(4,80),(4,82),(5,16),(5,28),(5,30),(5,34),(5,36),(5,39),(5,81),(5,82),(6,14),(6,21),(6,22),(6,25),(6,26),(6,45),(6,76),(6,82),(7,23),(7,24),(7,32),(7,41),(7,42),(7,45),(7,80),(7,81),(8,13),(8,16),(8,24),(8,26),(8,27),(8,43),(8,78),(8,79),(9,12),(9,15),(9,23),(9,25),(9,28),(9,44),(9,77),(9,79),(10,12),(10,17),(10,19),(10,30),(10,76),(10,78),(10,80),(11,13),(11,18),(11,20),(11,31),(11,76),(11,77),(11,81),(12,68),(12,101),(12,116),(12,118),(13,69),(13,102),(13,115),(13,119),(14,47),(14,89),(14,90),(14,103),(14,114),(15,53),(15,93),(15,99),(15,101),(15,104),(16,52),(16,94),(16,100),(16,102),(16,105),(17,62),(17,64),(17,91),(17,95),(17,118),(18,63),(18,65),(18,92),(18,96),(18,119),(19,60),(19,62),(19,90),(19,97),(19,116),(20,61),(20,63),(20,90),(20,98),(20,115),(21,56),(21,70),(21,87),(21,114),(21,118),(22,57),(22,71),(22,88),(22,114),(22,119),(23,73),(23,74),(23,101),(23,106),(23,120),(24,72),(24,75),(24,102),(24,106),(24,121),(25,71),(25,74),(25,103),(25,104),(25,118),(26,70),(26,75),(26,103),(26,105),(26,119),(27,66),(27,69),(27,99),(27,105),(27,121),(28,67),(28,68),(28,100),(28,104),(28,120),(29,46),(29,48),(29,49),(29,91),(29,92),(29,114),(30,60),(30,64),(30,68),(30,94),(30,122),(31,61),(31,65),(31,69),(31,93),(31,122),(32,47),(32,54),(32,55),(32,97),(32,98),(32,106),(33,50),(33,59),(33,61),(33,89),(33,97),(33,99),(34,51),(34,58),(34,60),(34,89),(34,98),(34,100),(35,48),(35,50),(35,66),(35,87),(35,93),(35,95),(36,49),(36,51),(36,67),(36,88),(36,94),(36,96),(37,50),(37,54),(37,58),(37,62),(37,114),(37,115),(38,51),(38,55),(38,59),(38,63),(38,114),(38,116),(39,49),(39,52),(39,58),(39,64),(39,87),(39,120),(40,48),(40,53),(40,59),(40,65),(40,88),(40,121),(41,46),(41,54),(41,56),(41,72),(41,95),(41,120),(42,46),(42,55),(42,57),(42,73),(42,96),(42,121),(43,52),(43,66),(43,70),(43,72),(43,91),(43,115),(44,53),(44,67),(44,71),(44,73),(44,92),(44,116),(45,47),(45,56),(45,57),(45,74),(45,75),(45,122),(46,123),(46,126),(46,127),(47,107),(47,123),(47,128),(48,111),(48,126),(48,131),(49,110),(49,127),(49,131),(50,112),(50,124),(50,131),(51,113),(51,125),(51,131),(52,108),(52,110),(52,129),(53,109),(53,111),(53,130),(54,112),(54,123),(54,129),(55,113),(55,123),(55,130),(56,85),(56,123),(56,132),(57,86),(57,123),(57,133),(58,83),(58,129),(58,131),(59,84),(59,130),(59,131),(60,83),(60,125),(60,128),(61,84),(61,124),(61,128),(62,83),(62,112),(62,134),(63,84),(63,113),(63,134),(64,83),(64,110),(64,132),(65,84),(65,111),(65,133),(66,108),(66,124),(66,126),(67,109),(67,125),(67,127),(68,125),(68,132),(69,124),(69,133),(70,85),(70,108),(70,134),(71,86),(71,109),(71,134),(72,85),(72,126),(72,129),(73,86),(73,127),(73,130),(74,86),(74,107),(74,132),(75,85),(75,107),(75,133),(76,90),(76,118),(76,119),(76,122),(77,92),(77,93),(77,115),(77,118),(77,120),(78,91),(78,94),(78,116),(78,119),(78,121),(79,99),(79,100),(79,103),(79,106),(79,115),(79,116),(80,95),(80,97),(80,101),(80,121),(80,122),(81,96),(81,98),(81,102),(81,120),(81,122),(82,87),(82,88),(82,89),(82,104),(82,105),(82,122),(83,135),(84,135),(85,135),(86,135),(87,108),(87,131),(87,132),(88,109),(88,131),(88,133),(89,117),(89,128),(89,131),(90,128),(90,134),(91,110),(91,126),(91,134),(92,111),(92,127),(92,134),(93,111),(93,124),(93,132),(94,110),(94,125),(94,133),(95,112),(95,126),(95,132),(96,113),(96,127),(96,133),(97,112),(97,128),(97,130),(98,113),(98,128),(98,129),(99,117),(99,124),(99,130),(100,117),(100,125),(100,129),(101,130),(101,132),(102,129),(102,133),(103,107),(103,117),(103,134),(104,109),(104,117),(104,132),(105,108),(105,117),(105,133),(106,107),(106,129),(106,130),(107,135),(108,135),(109,135),(110,135),(111,135),(112,135),(113,135),(114,123),(114,131),(114,134),(115,124),(115,129),(115,134),(116,125),(116,130),(116,134),(117,135),(118,132),(118,134),(119,133),(119,134),(120,127),(120,129),(120,132),(121,126),(121,130),(121,133),(122,128),(122,132),(122,133),(123,135),(124,135),(125,135),(126,135),(127,135),(128,135),(129,135),(130,135),(131,135),(132,135),(133,135),(134,135)],136)
=> ? = 3 - 1
[1,3,5,6,2,4] => [4,2,6,5,3,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,11),(1,12),(1,15),(1,58),(1,60),(1,61),(2,15),(2,24),(2,25),(2,26),(2,31),(2,62),(2,63),(3,17),(3,21),(3,28),(3,33),(3,37),(3,61),(3,63),(4,16),(4,20),(4,27),(4,32),(4,37),(4,60),(4,62),(5,18),(5,22),(5,30),(5,34),(5,36),(5,60),(5,63),(6,19),(6,23),(6,29),(6,35),(6,36),(6,61),(6,62),(7,31),(7,32),(7,33),(7,34),(7,35),(7,58),(7,59),(8,13),(8,14),(8,26),(8,27),(8,28),(8,29),(8,30),(8,58),(9,12),(9,14),(9,20),(9,21),(9,22),(9,23),(9,25),(9,59),(10,11),(10,13),(10,16),(10,17),(10,18),(10,19),(10,24),(10,59),(11,40),(11,68),(11,69),(11,94),(12,41),(12,70),(12,71),(12,94),(13,44),(13,45),(13,46),(13,47),(13,56),(13,94),(14,48),(14,49),(14,50),(14,51),(14,57),(14,94),(15,40),(15,41),(15,65),(15,97),(16,44),(16,52),(16,68),(16,72),(16,76),(17,45),(17,52),(17,69),(17,73),(17,77),(18,47),(18,53),(18,68),(18,74),(18,77),(19,46),(19,53),(19,69),(19,75),(19,76),(20,48),(20,54),(20,70),(20,72),(20,78),(21,49),(21,54),(21,71),(21,73),(21,79),(22,50),(22,55),(22,70),(22,74),(22,79),(23,51),(23,55),(23,71),(23,75),(23,78),(24,40),(24,56),(24,64),(24,76),(24,77),(25,41),(25,57),(25,64),(25,78),(25,79),(26,56),(26,57),(26,65),(26,80),(26,81),(27,38),(27,44),(27,48),(27,80),(27,92),(28,38),(28,45),(28,49),(28,81),(28,93),(29,39),(29,46),(29,51),(29,80),(29,93),(30,39),(30,47),(30,50),(30,81),(30,92),(31,64),(31,65),(31,66),(31,67),(32,42),(32,66),(32,72),(32,92),(33,42),(33,67),(33,73),(33,93),(34,43),(34,67),(34,74),(34,92),(35,43),(35,66),(35,75),(35,93),(36,39),(36,43),(36,53),(36,55),(36,97),(37,38),(37,42),(37,52),(37,54),(37,97),(38,82),(38,84),(38,103),(39,83),(39,85),(39,103),(40,98),(40,99),(41,98),(41,100),(42,86),(42,103),(43,87),(43,103),(44,82),(44,88),(44,101),(45,82),(45,89),(45,102),(46,83),(46,88),(46,102),(47,83),(47,89),(47,101),(48,84),(48,90),(48,101),(49,84),(49,91),(49,102),(50,85),(50,91),(50,101),(51,85),(51,90),(51,102),(52,82),(52,86),(52,99),(53,83),(53,87),(53,99),(54,84),(54,86),(54,100),(55,85),(55,87),(55,100),(56,88),(56,89),(56,98),(57,90),(57,91),(57,98),(58,65),(58,92),(58,93),(58,94),(59,64),(59,72),(59,73),(59,74),(59,75),(59,94),(60,68),(60,70),(60,92),(60,97),(61,69),(61,71),(61,93),(61,97),(62,66),(62,76),(62,78),(62,80),(62,97),(63,67),(63,77),(63,79),(63,81),(63,97),(64,95),(64,96),(64,98),(65,98),(65,103),(66,95),(66,103),(67,96),(67,103),(68,99),(68,101),(69,99),(69,102),(70,100),(70,101),(71,100),(71,102),(72,86),(72,95),(72,101),(73,86),(73,96),(73,102),(74,87),(74,96),(74,101),(75,87),(75,95),(75,102),(76,88),(76,95),(76,99),(77,89),(77,96),(77,99),(78,90),(78,95),(78,100),(79,91),(79,96),(79,100),(80,88),(80,90),(80,103),(81,89),(81,91),(81,103),(82,104),(83,104),(84,104),(85,104),(86,104),(87,104),(88,104),(89,104),(90,104),(91,104),(92,101),(92,103),(93,102),(93,103),(94,98),(94,101),(94,102),(95,104),(96,104),(97,99),(97,100),(97,103),(98,104),(99,104),(100,104),(101,104),(102,104),(103,104)],105)
=> ? = 3 - 1
[1,3,6,4,5,2] => [2,5,4,6,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,10),(1,22),(1,23),(1,24),(1,25),(1,26),(1,29),(1,30),(2,13),(2,18),(2,19),(2,20),(2,21),(2,30),(2,36),(3,12),(3,14),(3,15),(3,16),(3,17),(3,29),(3,36),(4,11),(4,12),(4,13),(4,26),(4,51),(4,52),(5,15),(5,19),(5,23),(5,28),(5,50),(5,52),(6,14),(6,18),(6,22),(6,28),(6,49),(6,51),(7,17),(7,20),(7,24),(7,27),(7,49),(7,52),(8,16),(8,21),(8,25),(8,27),(8,50),(8,51),(9,10),(9,11),(9,36),(9,49),(9,50),(10,31),(10,58),(10,59),(10,60),(11,31),(11,57),(11,74),(12,32),(12,57),(12,63),(12,64),(13,33),(13,57),(13,65),(13,66),(14,37),(14,45),(14,63),(14,72),(15,38),(15,45),(15,64),(15,73),(16,39),(16,46),(16,63),(16,73),(17,40),(17,46),(17,64),(17,72),(18,41),(18,47),(18,65),(18,72),(19,42),(19,47),(19,66),(19,73),(20,43),(20,48),(20,66),(20,72),(21,44),(21,48),(21,65),(21,73),(22,34),(22,37),(22,41),(22,58),(22,61),(23,34),(23,38),(23,42),(23,59),(23,62),(24,35),(24,40),(24,43),(24,58),(24,62),(25,35),(25,39),(25,44),(25,59),(25,61),(26,31),(26,32),(26,33),(26,61),(26,62),(27,35),(27,46),(27,48),(27,74),(28,34),(28,45),(28,47),(28,74),(29,32),(29,37),(29,38),(29,39),(29,40),(29,60),(30,33),(30,41),(30,42),(30,43),(30,44),(30,60),(31,71),(31,77),(32,67),(32,68),(32,71),(33,69),(33,70),(33,71),(34,53),(34,55),(34,77),(35,54),(35,56),(35,77),(36,57),(36,60),(36,72),(36,73),(37,53),(37,67),(37,75),(38,53),(38,68),(38,76),(39,54),(39,67),(39,76),(40,54),(40,68),(40,75),(41,55),(41,69),(41,75),(42,55),(42,70),(42,76),(43,56),(43,70),(43,75),(44,56),(44,69),(44,76),(45,53),(45,78),(46,54),(46,78),(47,55),(47,78),(48,56),(48,78),(49,58),(49,72),(49,74),(50,59),(50,73),(50,74),(51,61),(51,63),(51,65),(51,74),(52,62),(52,64),(52,66),(52,74),(53,79),(54,79),(55,79),(56,79),(57,71),(57,78),(58,75),(58,77),(59,76),(59,77),(60,71),(60,75),(60,76),(61,67),(61,69),(61,77),(62,68),(62,70),(62,77),(63,67),(63,78),(64,68),(64,78),(65,69),(65,78),(66,70),(66,78),(67,79),(68,79),(69,79),(70,79),(71,79),(72,75),(72,78),(73,76),(73,78),(74,77),(74,78),(75,79),(76,79),(77,79),(78,79)],80)
=> ? = 3 - 1
[1,3,6,5,2,4] => [4,2,5,6,3,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,20),(1,21),(1,22),(1,23),(1,33),(1,61),(2,10),(2,16),(2,17),(2,18),(2,19),(2,32),(2,61),(3,13),(3,17),(3,21),(3,26),(3,27),(3,29),(3,60),(4,12),(4,16),(4,20),(4,24),(4,25),(4,28),(4,60),(5,15),(5,19),(5,23),(5,25),(5,27),(5,31),(5,59),(6,14),(6,18),(6,22),(6,24),(6,26),(6,30),(6,59),(7,28),(7,29),(7,30),(7,31),(7,32),(7,33),(7,58),(8,12),(8,13),(8,14),(8,15),(8,58),(8,61),(9,10),(9,11),(9,58),(9,59),(9,60),(10,70),(10,71),(10,84),(11,72),(11,73),(11,84),(12,34),(12,35),(12,74),(12,82),(13,36),(13,37),(13,75),(13,82),(14,34),(14,36),(14,76),(14,83),(15,35),(15,37),(15,77),(15,83),(16,38),(16,39),(16,46),(16,70),(16,74),(17,40),(17,41),(17,47),(17,70),(17,75),(18,38),(18,40),(18,48),(18,71),(18,76),(19,39),(19,41),(19,49),(19,71),(19,77),(20,42),(20,43),(20,50),(20,72),(20,74),(21,44),(21,45),(21,51),(21,72),(21,75),(22,42),(22,44),(22,52),(22,73),(22,76),(23,43),(23,45),(23,53),(23,73),(23,77),(24,34),(24,38),(24,42),(24,54),(24,85),(25,35),(25,39),(25,43),(25,55),(25,85),(26,36),(26,40),(26,44),(26,56),(26,85),(27,37),(27,41),(27,45),(27,57),(27,85),(28,46),(28,50),(28,54),(28,55),(28,82),(29,47),(29,51),(29,56),(29,57),(29,82),(30,48),(30,52),(30,54),(30,56),(30,83),(31,49),(31,53),(31,55),(31,57),(31,83),(32,46),(32,47),(32,48),(32,49),(32,84),(33,50),(33,51),(33,52),(33,53),(33,84),(34,78),(34,90),(35,79),(35,90),(36,80),(36,90),(37,81),(37,90),(38,62),(38,78),(38,86),(39,63),(39,79),(39,86),(40,64),(40,80),(40,86),(41,65),(41,81),(41,86),(42,66),(42,78),(42,87),(43,67),(43,79),(43,87),(44,68),(44,80),(44,87),(45,69),(45,81),(45,87),(46,62),(46,63),(46,88),(47,64),(47,65),(47,88),(48,62),(48,64),(48,89),(49,63),(49,65),(49,89),(50,66),(50,67),(50,88),(51,68),(51,69),(51,88),(52,66),(52,68),(52,89),(53,67),(53,69),(53,89),(54,62),(54,66),(54,90),(55,63),(55,67),(55,90),(56,64),(56,68),(56,90),(57,65),(57,69),(57,90),(58,82),(58,83),(58,84),(59,71),(59,73),(59,83),(59,85),(60,70),(60,72),(60,82),(60,85),(61,74),(61,75),(61,76),(61,77),(61,84),(62,91),(63,91),(64,91),(65,91),(66,91),(67,91),(68,91),(69,91),(70,86),(70,88),(71,86),(71,89),(72,87),(72,88),(73,87),(73,89),(74,78),(74,79),(74,88),(75,80),(75,81),(75,88),(76,78),(76,80),(76,89),(77,79),(77,81),(77,89),(78,91),(79,91),(80,91),(81,91),(82,88),(82,90),(83,89),(83,90),(84,88),(84,89),(85,86),(85,87),(85,90),(86,91),(87,91),(88,91),(89,91),(90,91)],92)
=> ? = 3 - 1
[1,4,2,5,3,6] => [6,3,5,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,20),(1,29),(1,31),(1,36),(1,40),(1,42),(1,44),(1,81),(1,86),(2,19),(2,28),(2,30),(2,35),(2,39),(2,41),(2,43),(2,81),(2,85),(3,17),(3,26),(3,32),(3,35),(3,37),(3,42),(3,45),(3,82),(3,84),(4,18),(4,27),(4,33),(4,36),(4,38),(4,41),(4,46),(4,82),(4,83),(5,25),(5,37),(5,38),(5,39),(5,40),(5,47),(5,48),(5,79),(5,80),(6,15),(6,16),(6,24),(6,32),(6,33),(6,34),(6,48),(6,85),(6,86),(7,13),(7,14),(7,23),(7,30),(7,31),(7,34),(7,47),(7,83),(7,84),(8,13),(8,17),(8,22),(8,27),(8,43),(8,49),(8,79),(8,86),(9,14),(9,18),(9,21),(9,26),(9,44),(9,49),(9,80),(9,85),(10,15),(10,19),(10,21),(10,29),(10,45),(10,50),(10,79),(10,83),(11,16),(11,20),(11,22),(11,28),(11,46),(11,50),(11,80),(11,84),(12,23),(12,24),(12,25),(12,49),(12,50),(12,81),(12,82),(13,63),(13,91),(13,103),(13,108),(13,127),(14,64),(14,90),(14,102),(14,108),(14,128),(15,65),(15,92),(15,104),(15,109),(15,130),(16,66),(16,93),(16,105),(16,109),(16,129),(17,59),(17,96),(17,101),(17,103),(17,131),(18,60),(18,97),(18,100),(18,102),(18,131),(19,61),(19,94),(19,98),(19,104),(19,132),(20,62),(20,95),(20,99),(20,105),(20,132),(21,73),(21,74),(21,102),(21,104),(21,138),(22,72),(22,75),(22,103),(22,105),(22,138),(23,76),(23,78),(23,108),(23,110),(23,139),(24,77),(24,78),(24,109),(24,111),(24,140),(25,76),(25,77),(25,106),(25,107),(25,138),(26,69),(26,74),(26,114),(26,128),(26,131),(27,68),(27,75),(27,115),(27,127),(27,131),(28,70),(28,72),(28,112),(28,129),(28,132),(29,71),(29,73),(29,113),(29,130),(29,132),(30,53),(30,63),(30,90),(30,98),(30,110),(30,112),(31,54),(31,64),(31,91),(31,99),(31,110),(31,113),(32,56),(32,65),(32,93),(32,101),(32,111),(32,114),(33,55),(33,66),(33,92),(33,100),(33,111),(33,115),(34,67),(34,78),(34,90),(34,91),(34,92),(34,93),(35,51),(35,59),(35,61),(35,112),(35,114),(35,124),(36,52),(36,60),(36,62),(36,113),(36,115),(36,124),(37,51),(37,56),(37,58),(37,96),(37,106),(37,128),(38,52),(38,55),(38,57),(38,97),(38,106),(38,127),(39,51),(39,53),(39,57),(39,94),(39,107),(39,129),(40,52),(40,54),(40,58),(40,95),(40,107),(40,130),(41,57),(41,68),(41,70),(41,98),(41,100),(41,124),(42,58),(42,69),(42,71),(42,99),(42,101),(42,124),(43,59),(43,63),(43,68),(43,72),(43,94),(43,140),(44,60),(44,64),(44,69),(44,73),(44,95),(44,140),(45,61),(45,65),(45,71),(45,74),(45,96),(45,139),(46,62),(46,66),(46,70),(46,75),(46,97),(46,139),(47,53),(47,54),(47,67),(47,76),(47,127),(47,128),(48,55),(48,56),(48,67),(48,77),(48,129),(48,130),(49,108),(49,131),(49,138),(49,140),(50,109),(50,132),(50,138),(50,139),(51,122),(51,133),(51,143),(52,123),(52,133),(52,144),(53,116),(53,134),(53,143),(54,116),(54,135),(54,144),(55,117),(55,136),(55,144),(56,117),(56,137),(56,143),(57,133),(57,134),(57,136),(58,133),(58,135),(58,137),(59,118),(59,122),(59,146),(60,119),(60,123),(60,146),(61,120),(61,122),(61,145),(62,121),(62,123),(62,145),(63,118),(63,134),(63,141),(64,119),(64,135),(64,141),(65,120),(65,137),(65,142),(66,121),(66,136),(66,142),(67,87),(67,143),(67,144),(68,88),(68,134),(68,146),(69,89),(69,135),(69,146),(70,88),(70,136),(70,145),(71,89),(71,137),(71,145),(72,88),(72,118),(72,148),(73,89),(73,119),(73,148),(74,89),(74,120),(74,147),(75,88),(75,121),(75,147),(76,87),(76,116),(76,147),(77,87),(77,117),(77,148),(78,87),(78,141),(78,142),(79,94),(79,96),(79,127),(79,130),(79,138),(80,95),(80,97),(80,128),(80,129),(80,138),(81,107),(81,110),(81,124),(81,132),(81,140),(82,106),(82,111),(82,124),(82,131),(82,139),(83,92),(83,98),(83,102),(83,113),(83,127),(83,139),(84,93),(84,99),(84,103),(84,112),(84,128),(84,139),(85,90),(85,100),(85,104),(85,114),(85,129),(85,140),(86,91),(86,101),(86,105),(86,115),(86,130),(86,140),(87,149),(88,149),(89,149),(90,125),(90,141),(90,143),(91,126),(91,141),(91,144),(92,125),(92,142),(92,144),(93,126),(93,142),(93,143),(94,122),(94,134),(94,148),(95,123),(95,135),(95,148),(96,122),(96,137),(96,147),(97,123),(97,136),(97,147),(98,125),(98,134),(98,145),(99,126),(99,135),(99,145),(100,125),(100,136),(100,146),(101,126),(101,137),(101,146),(102,119),(102,125),(102,147),(103,118),(103,126),(103,147),(104,120),(104,125),(104,148),(105,121),(105,126),(105,148),(106,117),(106,133),(106,147),(107,116),(107,133),(107,148),(108,141),(108,147),(109,142),(109,148),(110,116),(110,141),(110,145),(111,117),(111,142),(111,146),(112,118),(112,143),(112,145),(113,119),(113,144),(113,145),(114,120),(114,143),(114,146),(115,121),(115,144),(115,146),(116,149),(117,149),(118,149),(119,149),(120,149),(121,149),(122,149),(123,149),(124,133),(124,145),(124,146),(125,149),(126,149),(127,134),(127,144),(127,147),(128,135),(128,143),(128,147),(129,136),(129,143),(129,148),(130,137),(130,144),(130,148),(131,146),(131,147),(132,145),(132,148),(133,149),(134,149),(135,149),(136,149),(137,149),(138,147),(138,148),(139,142),(139,145),(139,147),(140,141),(140,146),(140,148),(141,149),(142,149),(143,149),(144,149),(145,149),(146,149),(147,149),(148,149)],150)
=> ? = 3 - 1
[1,4,2,5,6,3] => [3,6,5,2,4,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,16),(1,20),(1,21),(1,32),(1,35),(1,36),(1,41),(1,42),(1,43),(2,15),(2,18),(2,19),(2,31),(2,33),(2,34),(2,39),(2,40),(2,43),(3,23),(3,25),(3,28),(3,38),(3,40),(3,42),(3,71),(3,75),(4,22),(4,24),(4,27),(4,37),(4,39),(4,41),(4,70),(4,75),(5,17),(5,22),(5,23),(5,26),(5,31),(5,32),(5,72),(5,73),(6,13),(6,24),(6,29),(6,33),(6,35),(6,38),(6,72),(6,74),(7,12),(7,25),(7,30),(7,34),(7,36),(7,37),(7,73),(7,74),(8,12),(8,13),(8,14),(8,17),(8,43),(8,70),(8,71),(9,14),(9,15),(9,16),(9,26),(9,44),(9,74),(9,75),(10,18),(10,20),(10,27),(10,30),(10,44),(10,71),(10,72),(11,19),(11,21),(11,28),(11,29),(11,44),(11,70),(11,73),(12,80),(12,88),(12,94),(12,108),(13,80),(13,87),(13,95),(13,109),(14,45),(14,80),(14,83),(14,117),(15,62),(15,81),(15,83),(15,84),(15,92),(16,63),(16,82),(16,83),(16,85),(16,93),(17,45),(17,91),(17,108),(17,109),(18,58),(18,65),(18,84),(18,96),(18,111),(19,59),(19,64),(19,84),(19,97),(19,110),(20,60),(20,67),(20,85),(20,99),(20,111),(21,61),(21,66),(21,85),(21,98),(21,110),(22,46),(22,48),(22,86),(22,89),(22,108),(23,47),(23,49),(23,86),(23,90),(23,109),(24,50),(24,52),(24,87),(24,89),(24,107),(25,51),(25,53),(25,88),(25,90),(25,107),(26,45),(26,62),(26,63),(26,86),(26,118),(27,58),(27,60),(27,68),(27,89),(27,117),(28,59),(28,61),(28,69),(28,90),(28,117),(29,64),(29,66),(29,69),(29,87),(29,118),(30,65),(30,67),(30,68),(30,88),(30,118),(31,46),(31,47),(31,62),(31,91),(31,96),(31,97),(32,48),(32,49),(32,63),(32,91),(32,98),(32,99),(33,50),(33,55),(33,64),(33,92),(33,95),(33,96),(34,51),(34,54),(34,65),(34,92),(34,94),(34,97),(35,52),(35,57),(35,66),(35,93),(35,95),(35,99),(36,53),(36,56),(36,67),(36,93),(36,94),(36,98),(37,54),(37,56),(37,68),(37,107),(37,108),(38,55),(38,57),(38,69),(38,107),(38,109),(39,46),(39,50),(39,54),(39,58),(39,81),(39,110),(40,47),(40,51),(40,55),(40,59),(40,81),(40,111),(41,48),(41,52),(41,56),(41,60),(41,82),(41,110),(42,49),(42,53),(42,57),(42,61),(42,82),(42,111),(43,83),(43,91),(43,94),(43,95),(43,110),(43,111),(44,84),(44,85),(44,117),(44,118),(45,106),(45,124),(46,100),(46,104),(46,121),(47,101),(47,104),(47,122),(48,102),(48,105),(48,121),(49,103),(49,105),(49,122),(50,100),(50,113),(50,115),(51,101),(51,114),(51,115),(52,102),(52,113),(52,116),(53,103),(53,114),(53,116),(54,76),(54,115),(54,121),(55,77),(55,115),(55,122),(56,78),(56,116),(56,121),(57,79),(57,116),(57,122),(58,76),(58,100),(58,123),(59,77),(59,101),(59,123),(60,78),(60,102),(60,123),(61,79),(61,103),(61,123),(62,104),(62,106),(62,119),(63,105),(63,106),(63,120),(64,77),(64,113),(64,119),(65,76),(65,114),(65,119),(66,79),(66,113),(66,120),(67,78),(67,114),(67,120),(68,76),(68,78),(68,124),(69,77),(69,79),(69,124),(70,87),(70,108),(70,110),(70,117),(71,88),(71,109),(71,111),(71,117),(72,89),(72,96),(72,99),(72,109),(72,118),(73,90),(73,97),(73,98),(73,108),(73,118),(74,80),(74,92),(74,93),(74,107),(74,118),(75,81),(75,82),(75,86),(75,107),(75,117),(76,125),(77,125),(78,125),(79,125),(80,112),(80,124),(81,104),(81,115),(81,123),(82,105),(82,116),(82,123),(83,106),(83,112),(83,123),(84,119),(84,123),(85,120),(85,123),(86,104),(86,105),(86,124),(87,113),(87,124),(88,114),(88,124),(89,100),(89,102),(89,124),(90,101),(90,103),(90,124),(91,106),(91,121),(91,122),(92,112),(92,115),(92,119),(93,112),(93,116),(93,120),(94,112),(94,114),(94,121),(95,112),(95,113),(95,122),(96,100),(96,119),(96,122),(97,101),(97,119),(97,121),(98,103),(98,120),(98,121),(99,102),(99,120),(99,122),(100,125),(101,125),(102,125),(103,125),(104,125),(105,125),(106,125),(107,115),(107,116),(107,124),(108,121),(108,124),(109,122),(109,124),(110,113),(110,121),(110,123),(111,114),(111,122),(111,123),(112,125),(113,125),(114,125),(115,125),(116,125),(117,123),(117,124),(118,119),(118,120),(118,124),(119,125),(120,125),(121,125),(122,125),(123,125),(124,125)],126)
=> ? = 3 - 1
[1,4,2,6,5,3] => [3,5,6,2,4,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,18),(1,28),(1,30),(1,32),(1,35),(1,36),(1,72),(2,12),(2,17),(2,27),(2,29),(2,32),(2,33),(2,34),(2,71),(3,17),(3,20),(3,21),(3,30),(3,31),(3,37),(3,38),(3,73),(4,18),(4,19),(4,22),(4,29),(4,31),(4,39),(4,40),(4,74),(5,16),(5,24),(5,26),(5,34),(5,36),(5,38),(5,40),(5,75),(6,15),(6,23),(6,25),(6,33),(6,35),(6,37),(6,39),(6,75),(7,11),(7,15),(7,16),(7,19),(7,20),(7,71),(7,72),(8,14),(8,22),(8,23),(8,24),(8,27),(8,72),(8,73),(9,14),(9,21),(9,25),(9,26),(9,28),(9,71),(9,74),(10,11),(10,12),(10,13),(10,73),(10,74),(10,75),(11,84),(11,99),(11,100),(12,41),(12,82),(12,85),(12,99),(13,41),(13,83),(13,86),(13,100),(14,58),(14,59),(14,101),(14,102),(15,54),(15,56),(15,84),(15,87),(15,89),(16,55),(16,57),(16,84),(16,88),(16,90),(17,44),(17,45),(17,85),(17,91),(17,98),(18,46),(18,47),(18,86),(18,92),(18,98),(19,54),(19,55),(19,68),(19,92),(19,99),(20,56),(20,57),(20,68),(20,91),(20,100),(21,64),(21,65),(21,70),(21,91),(21,102),(22,66),(22,67),(22,69),(22,92),(22,102),(23,58),(23,60),(23,66),(23,89),(23,103),(24,59),(24,61),(24,67),(24,90),(24,103),(25,58),(25,62),(25,64),(25,87),(25,104),(26,59),(26,63),(26,65),(26,88),(26,104),(27,60),(27,61),(27,69),(27,85),(27,101),(28,62),(28,63),(28,70),(28,86),(28,101),(29,50),(29,51),(29,69),(29,98),(29,99),(30,52),(30,53),(30,70),(30,98),(30,100),(31,48),(31,49),(31,68),(31,98),(31,102),(32,41),(32,42),(32,43),(32,98),(32,101),(33,42),(33,44),(33,50),(33,60),(33,82),(33,87),(34,43),(34,45),(34,51),(34,61),(34,82),(34,88),(35,42),(35,46),(35,52),(35,62),(35,83),(35,89),(36,43),(36,47),(36,53),(36,63),(36,83),(36,90),(37,44),(37,48),(37,52),(37,56),(37,64),(37,103),(38,45),(38,49),(38,53),(38,57),(38,65),(38,103),(39,46),(39,48),(39,50),(39,54),(39,66),(39,104),(40,47),(40,49),(40,51),(40,55),(40,67),(40,104),(41,97),(41,114),(42,97),(42,105),(42,107),(43,97),(43,106),(43,108),(44,93),(44,105),(44,109),(45,94),(45,106),(45,109),(46,95),(46,105),(46,110),(47,96),(47,106),(47,110),(48,80),(48,105),(48,113),(49,81),(49,106),(49,113),(50,76),(50,105),(50,111),(51,77),(51,106),(51,111),(52,78),(52,105),(52,112),(53,79),(53,106),(53,112),(54,80),(54,95),(54,111),(55,81),(55,96),(55,111),(56,80),(56,93),(56,112),(57,81),(57,94),(57,112),(58,107),(58,113),(59,108),(59,113),(60,76),(60,107),(60,109),(61,77),(61,108),(61,109),(62,78),(62,107),(62,110),(63,79),(63,108),(63,110),(64,78),(64,93),(64,113),(65,79),(65,94),(65,113),(66,76),(66,95),(66,113),(67,77),(67,96),(67,113),(68,80),(68,81),(68,114),(69,76),(69,77),(69,114),(70,78),(70,79),(70,114),(71,87),(71,88),(71,91),(71,99),(71,101),(72,89),(72,90),(72,92),(72,100),(72,101),(73,85),(73,100),(73,102),(73,103),(74,86),(74,99),(74,102),(74,104),(75,82),(75,83),(75,84),(75,103),(75,104),(76,115),(77,115),(78,115),(79,115),(80,115),(81,115),(82,97),(82,109),(82,111),(83,97),(83,110),(83,112),(84,111),(84,112),(85,109),(85,114),(86,110),(86,114),(87,93),(87,107),(87,111),(88,94),(88,108),(88,111),(89,95),(89,107),(89,112),(90,96),(90,108),(90,112),(91,93),(91,94),(91,114),(92,95),(92,96),(92,114),(93,115),(94,115),(95,115),(96,115),(97,115),(98,105),(98,106),(98,114),(99,111),(99,114),(100,112),(100,114),(101,107),(101,108),(101,114),(102,113),(102,114),(103,109),(103,112),(103,113),(104,110),(104,111),(104,113),(105,115),(106,115),(107,115),(108,115),(109,115),(110,115),(111,115),(112,115),(113,115),(114,115)],116)
=> ? = 3 - 1
[1,4,5,2,3,6] => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,12),(1,13),(1,64),(1,65),(1,66),(1,67),(2,15),(2,24),(2,28),(2,32),(2,36),(2,42),(2,65),(2,71),(3,14),(3,23),(3,27),(3,31),(3,35),(3,42),(3,64),(3,70),(4,17),(4,26),(4,30),(4,33),(4,37),(4,41),(4,64),(4,71),(5,16),(5,25),(5,29),(5,34),(5,38),(5,41),(5,65),(5,70),(6,19),(6,27),(6,28),(6,29),(6,30),(6,40),(6,67),(6,69),(7,18),(7,23),(7,24),(7,25),(7,26),(7,40),(7,66),(7,68),(8,21),(8,35),(8,36),(8,37),(8,38),(8,39),(8,66),(8,69),(9,20),(9,31),(9,32),(9,33),(9,34),(9,39),(9,67),(9,68),(10,13),(10,18),(10,19),(10,20),(10,21),(10,22),(10,70),(10,71),(11,12),(11,14),(11,15),(11,16),(11,17),(11,22),(11,68),(11,69),(12,43),(12,76),(12,77),(12,116),(13,43),(13,78),(13,79),(13,117),(14,60),(14,72),(14,76),(14,80),(14,84),(15,60),(15,73),(15,77),(15,81),(15,85),(16,61),(16,72),(16,77),(16,82),(16,86),(17,61),(17,73),(17,76),(17,83),(17,87),(18,62),(18,74),(18,78),(18,88),(18,89),(19,62),(19,75),(19,79),(19,90),(19,91),(20,63),(20,74),(20,79),(20,92),(20,93),(21,63),(21,75),(21,78),(21,94),(21,95),(22,43),(22,72),(22,73),(22,74),(22,75),(23,44),(23,52),(23,80),(23,88),(23,108),(24,45),(24,52),(24,81),(24,89),(24,109),(25,47),(25,53),(25,82),(25,88),(25,109),(26,46),(26,53),(26,83),(26,89),(26,108),(27,44),(27,54),(27,84),(27,90),(27,110),(28,45),(28,54),(28,85),(28,91),(28,111),(29,47),(29,55),(29,86),(29,90),(29,111),(30,46),(30,55),(30,87),(30,91),(30,110),(31,48),(31,58),(31,80),(31,92),(31,110),(32,49),(32,58),(32,81),(32,93),(32,111),(33,50),(33,59),(33,83),(33,93),(33,110),(34,51),(34,59),(34,82),(34,92),(34,111),(35,48),(35,56),(35,84),(35,94),(35,108),(36,49),(36,56),(36,85),(36,95),(36,109),(37,50),(37,57),(37,87),(37,95),(37,108),(38,51),(38,57),(38,86),(38,94),(38,109),(39,48),(39,49),(39,50),(39,51),(39,63),(39,116),(40,44),(40,45),(40,46),(40,47),(40,62),(40,116),(41,53),(41,55),(41,57),(41,59),(41,61),(41,117),(42,52),(42,54),(42,56),(42,58),(42,60),(42,117),(43,118),(43,119),(44,96),(44,104),(44,120),(45,96),(45,105),(45,121),(46,97),(46,105),(46,120),(47,97),(47,104),(47,121),(48,98),(48,106),(48,120),(49,98),(49,107),(49,121),(50,99),(50,107),(50,120),(51,99),(51,106),(51,121),(52,96),(52,100),(52,122),(53,97),(53,101),(53,122),(54,96),(54,102),(54,123),(55,97),(55,103),(55,123),(56,98),(56,102),(56,122),(57,99),(57,103),(57,122),(58,98),(58,100),(58,123),(59,99),(59,101),(59,123),(60,100),(60,102),(60,118),(61,101),(61,103),(61,118),(62,104),(62,105),(62,119),(63,106),(63,107),(63,119),(64,76),(64,108),(64,110),(64,117),(65,77),(65,109),(65,111),(65,117),(66,78),(66,108),(66,109),(66,116),(67,79),(67,110),(67,111),(67,116),(68,74),(68,80),(68,81),(68,82),(68,83),(68,116),(69,75),(69,84),(69,85),(69,86),(69,87),(69,116),(70,72),(70,88),(70,90),(70,92),(70,94),(70,117),(71,73),(71,89),(71,91),(71,93),(71,95),(71,117),(72,112),(72,114),(72,118),(73,113),(73,115),(73,118),(74,112),(74,113),(74,119),(75,114),(75,115),(75,119),(76,118),(76,120),(77,118),(77,121),(78,119),(78,122),(79,119),(79,123),(80,100),(80,112),(80,120),(81,100),(81,113),(81,121),(82,101),(82,112),(82,121),(83,101),(83,113),(83,120),(84,102),(84,114),(84,120),(85,102),(85,115),(85,121),(86,103),(86,114),(86,121),(87,103),(87,115),(87,120),(88,104),(88,112),(88,122),(89,105),(89,113),(89,122),(90,104),(90,114),(90,123),(91,105),(91,115),(91,123),(92,106),(92,112),(92,123),(93,107),(93,113),(93,123),(94,106),(94,114),(94,122),(95,107),(95,115),(95,122),(96,124),(97,124),(98,124),(99,124),(100,124),(101,124),(102,124),(103,124),(104,124),(105,124),(106,124),(107,124),(108,120),(108,122),(109,121),(109,122),(110,120),(110,123),(111,121),(111,123),(112,124),(113,124),(114,124),(115,124),(116,119),(116,120),(116,121),(117,118),(117,122),(117,123),(118,124),(119,124),(120,124),(121,124),(122,124),(123,124)],125)
=> ? = 3 - 1
[1,4,5,2,6,3] => [3,6,2,5,4,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,11),(1,12),(1,15),(1,58),(1,60),(1,61),(2,15),(2,24),(2,25),(2,26),(2,31),(2,62),(2,63),(3,17),(3,21),(3,28),(3,33),(3,37),(3,61),(3,63),(4,16),(4,20),(4,27),(4,32),(4,37),(4,60),(4,62),(5,18),(5,22),(5,30),(5,34),(5,36),(5,60),(5,63),(6,19),(6,23),(6,29),(6,35),(6,36),(6,61),(6,62),(7,31),(7,32),(7,33),(7,34),(7,35),(7,58),(7,59),(8,13),(8,14),(8,26),(8,27),(8,28),(8,29),(8,30),(8,58),(9,12),(9,14),(9,20),(9,21),(9,22),(9,23),(9,25),(9,59),(10,11),(10,13),(10,16),(10,17),(10,18),(10,19),(10,24),(10,59),(11,40),(11,68),(11,69),(11,94),(12,41),(12,70),(12,71),(12,94),(13,44),(13,45),(13,46),(13,47),(13,56),(13,94),(14,48),(14,49),(14,50),(14,51),(14,57),(14,94),(15,40),(15,41),(15,65),(15,97),(16,44),(16,52),(16,68),(16,72),(16,76),(17,45),(17,52),(17,69),(17,73),(17,77),(18,47),(18,53),(18,68),(18,74),(18,77),(19,46),(19,53),(19,69),(19,75),(19,76),(20,48),(20,54),(20,70),(20,72),(20,78),(21,49),(21,54),(21,71),(21,73),(21,79),(22,50),(22,55),(22,70),(22,74),(22,79),(23,51),(23,55),(23,71),(23,75),(23,78),(24,40),(24,56),(24,64),(24,76),(24,77),(25,41),(25,57),(25,64),(25,78),(25,79),(26,56),(26,57),(26,65),(26,80),(26,81),(27,38),(27,44),(27,48),(27,80),(27,92),(28,38),(28,45),(28,49),(28,81),(28,93),(29,39),(29,46),(29,51),(29,80),(29,93),(30,39),(30,47),(30,50),(30,81),(30,92),(31,64),(31,65),(31,66),(31,67),(32,42),(32,66),(32,72),(32,92),(33,42),(33,67),(33,73),(33,93),(34,43),(34,67),(34,74),(34,92),(35,43),(35,66),(35,75),(35,93),(36,39),(36,43),(36,53),(36,55),(36,97),(37,38),(37,42),(37,52),(37,54),(37,97),(38,82),(38,84),(38,103),(39,83),(39,85),(39,103),(40,98),(40,99),(41,98),(41,100),(42,86),(42,103),(43,87),(43,103),(44,82),(44,88),(44,101),(45,82),(45,89),(45,102),(46,83),(46,88),(46,102),(47,83),(47,89),(47,101),(48,84),(48,90),(48,101),(49,84),(49,91),(49,102),(50,85),(50,91),(50,101),(51,85),(51,90),(51,102),(52,82),(52,86),(52,99),(53,83),(53,87),(53,99),(54,84),(54,86),(54,100),(55,85),(55,87),(55,100),(56,88),(56,89),(56,98),(57,90),(57,91),(57,98),(58,65),(58,92),(58,93),(58,94),(59,64),(59,72),(59,73),(59,74),(59,75),(59,94),(60,68),(60,70),(60,92),(60,97),(61,69),(61,71),(61,93),(61,97),(62,66),(62,76),(62,78),(62,80),(62,97),(63,67),(63,77),(63,79),(63,81),(63,97),(64,95),(64,96),(64,98),(65,98),(65,103),(66,95),(66,103),(67,96),(67,103),(68,99),(68,101),(69,99),(69,102),(70,100),(70,101),(71,100),(71,102),(72,86),(72,95),(72,101),(73,86),(73,96),(73,102),(74,87),(74,96),(74,101),(75,87),(75,95),(75,102),(76,88),(76,95),(76,99),(77,89),(77,96),(77,99),(78,90),(78,95),(78,100),(79,91),(79,96),(79,100),(80,88),(80,90),(80,103),(81,89),(81,91),(81,103),(82,104),(83,104),(84,104),(85,104),(86,104),(87,104),(88,104),(89,104),(90,104),(91,104),(92,101),(92,103),(93,102),(93,103),(94,98),(94,101),(94,102),(95,104),(96,104),(97,99),(97,100),(97,103),(98,104),(99,104),(100,104),(101,104),(102,104),(103,104)],105)
=> ? = 3 - 1
[1,4,5,6,2,3] => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,17),(1,23),(1,30),(1,42),(1,44),(2,10),(2,16),(2,22),(2,30),(2,41),(2,43),(3,12),(3,18),(3,24),(3,29),(3,41),(3,44),(4,13),(4,19),(4,25),(4,29),(4,42),(4,43),(5,15),(5,21),(5,27),(5,28),(5,43),(5,44),(6,14),(6,20),(6,26),(6,28),(6,41),(6,42),(7,22),(7,23),(7,24),(7,25),(7,26),(7,27),(7,31),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,31),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,31),(10,32),(10,45),(10,47),(10,57),(11,32),(11,46),(11,48),(11,58),(12,33),(12,45),(12,48),(12,59),(13,33),(13,46),(13,47),(13,60),(14,34),(14,45),(14,46),(14,61),(15,34),(15,47),(15,48),(15,62),(16,35),(16,49),(16,51),(16,57),(17,35),(17,50),(17,52),(17,58),(18,36),(18,49),(18,52),(18,59),(19,36),(19,50),(19,51),(19,60),(20,37),(20,49),(20,50),(20,61),(21,37),(21,51),(21,52),(21,62),(22,38),(22,53),(22,55),(22,57),(23,38),(23,54),(23,56),(23,58),(24,39),(24,53),(24,56),(24,59),(25,39),(25,54),(25,55),(25,60),(26,40),(26,53),(26,54),(26,61),(27,40),(27,55),(27,56),(27,62),(28,34),(28,37),(28,40),(28,70),(29,33),(29,36),(29,39),(29,70),(30,32),(30,35),(30,38),(30,70),(31,57),(31,58),(31,59),(31,60),(31,61),(31,62),(32,63),(32,71),(33,64),(33,71),(34,65),(34,71),(35,63),(35,72),(36,64),(36,72),(37,65),(37,72),(38,63),(38,73),(39,64),(39,73),(40,65),(40,73),(41,45),(41,49),(41,53),(41,70),(42,46),(42,50),(42,54),(42,70),(43,47),(43,51),(43,55),(43,70),(44,48),(44,52),(44,56),(44,70),(45,66),(45,71),(46,67),(46,71),(47,68),(47,71),(48,69),(48,71),(49,66),(49,72),(50,67),(50,72),(51,68),(51,72),(52,69),(52,72),(53,66),(53,73),(54,67),(54,73),(55,68),(55,73),(56,69),(56,73),(57,63),(57,66),(57,68),(58,63),(58,67),(58,69),(59,64),(59,66),(59,69),(60,64),(60,67),(60,68),(61,65),(61,66),(61,67),(62,65),(62,68),(62,69),(63,74),(64,74),(65,74),(66,74),(67,74),(68,74),(69,74),(70,71),(70,72),(70,73),(71,74),(72,74),(73,74)],75)
=> ? = 3 - 1
[1,4,6,2,5,3] => [3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,15),(1,16),(1,17),(1,18),(1,54),(1,55),(2,10),(2,13),(2,14),(2,20),(2,55),(2,57),(3,10),(3,11),(3,12),(3,19),(3,54),(3,56),(4,12),(4,16),(4,23),(4,24),(4,26),(4,30),(4,57),(5,11),(5,15),(5,21),(5,22),(5,25),(5,29),(5,57),(6,14),(6,18),(6,22),(6,24),(6,28),(6,32),(6,56),(7,13),(7,17),(7,21),(7,23),(7,27),(7,31),(7,56),(8,19),(8,27),(8,28),(8,29),(8,30),(8,33),(8,55),(9,20),(9,25),(9,26),(9,31),(9,32),(9,33),(9,54),(10,58),(10,59),(10,78),(11,42),(11,58),(11,60),(11,64),(12,43),(12,58),(12,61),(12,65),(13,44),(13,59),(13,62),(13,66),(14,45),(14,59),(14,63),(14,67),(15,38),(15,39),(15,60),(15,80),(16,40),(16,41),(16,61),(16,80),(17,38),(17,40),(17,62),(17,79),(18,39),(18,41),(18,63),(18,79),(19,42),(19,43),(19,68),(19,78),(20,44),(20,45),(20,69),(20,78),(21,38),(21,46),(21,50),(21,64),(21,66),(22,39),(22,47),(22,51),(22,64),(22,67),(23,40),(23,48),(23,52),(23,65),(23,66),(24,41),(24,49),(24,53),(24,65),(24,67),(25,34),(25,50),(25,51),(25,60),(25,69),(26,35),(26,52),(26,53),(26,61),(26,69),(27,36),(27,46),(27,48),(27,62),(27,68),(28,37),(28,47),(28,49),(28,63),(28,68),(29,34),(29,42),(29,46),(29,47),(29,80),(30,35),(30,43),(30,48),(30,49),(30,80),(31,36),(31,44),(31,50),(31,52),(31,79),(32,37),(32,45),(32,51),(32,53),(32,79),(33,34),(33,35),(33,36),(33,37),(33,78),(34,70),(34,71),(34,86),(35,72),(35,73),(35,86),(36,70),(36,72),(36,87),(37,71),(37,73),(37,87),(38,82),(38,84),(39,82),(39,85),(40,83),(40,84),(41,83),(41,85),(42,74),(42,86),(43,75),(43,86),(44,76),(44,87),(45,77),(45,87),(46,70),(46,74),(46,84),(47,71),(47,74),(47,85),(48,72),(48,75),(48,84),(49,73),(49,75),(49,85),(50,70),(50,76),(50,82),(51,71),(51,77),(51,82),(52,72),(52,76),(52,83),(53,73),(53,77),(53,83),(54,60),(54,61),(54,78),(54,79),(55,62),(55,63),(55,78),(55,80),(56,59),(56,64),(56,65),(56,68),(56,79),(57,58),(57,66),(57,67),(57,69),(57,80),(58,81),(58,86),(59,81),(59,87),(60,82),(60,86),(61,83),(61,86),(62,84),(62,87),(63,85),(63,87),(64,74),(64,81),(64,82),(65,75),(65,81),(65,83),(66,76),(66,81),(66,84),(67,77),(67,81),(67,85),(68,74),(68,75),(68,87),(69,76),(69,77),(69,86),(70,88),(71,88),(72,88),(73,88),(74,88),(75,88),(76,88),(77,88),(78,86),(78,87),(79,82),(79,83),(79,87),(80,84),(80,85),(80,86),(81,88),(82,88),(83,88),(84,88),(85,88),(86,88),(87,88)],89)
=> ? = 3 - 1
[1,4,6,5,2,3] => [3,2,5,6,4,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,10),(1,11),(1,40),(1,41),(2,15),(2,16),(2,17),(2,26),(2,27),(2,41),(3,12),(3,13),(3,14),(3,24),(3,25),(3,41),(4,21),(4,22),(4,23),(4,25),(4,27),(4,40),(5,18),(5,19),(5,20),(5,24),(5,26),(5,40),(6,11),(6,14),(6,17),(6,20),(6,23),(6,42),(7,10),(7,13),(7,16),(7,19),(7,22),(7,42),(8,9),(8,12),(8,15),(8,18),(8,21),(8,42),(9,43),(9,46),(9,49),(10,44),(10,47),(10,49),(11,45),(11,48),(11,49),(12,28),(12,31),(12,43),(12,50),(13,29),(13,32),(13,44),(13,50),(14,30),(14,33),(14,45),(14,50),(15,34),(15,37),(15,43),(15,51),(16,35),(16,38),(16,44),(16,51),(17,36),(17,39),(17,45),(17,51),(18,28),(18,34),(18,46),(18,52),(19,29),(19,35),(19,47),(19,52),(20,30),(20,36),(20,48),(20,52),(21,31),(21,37),(21,46),(21,53),(22,32),(22,38),(22,47),(22,53),(23,33),(23,39),(23,48),(23,53),(24,28),(24,29),(24,30),(24,58),(25,31),(25,32),(25,33),(25,58),(26,34),(26,35),(26,36),(26,58),(27,37),(27,38),(27,39),(27,58),(28,54),(28,61),(29,54),(29,62),(30,54),(30,63),(31,55),(31,61),(32,55),(32,62),(33,55),(33,63),(34,56),(34,61),(35,56),(35,62),(36,56),(36,63),(37,57),(37,61),(38,57),(38,62),(39,57),(39,63),(40,46),(40,47),(40,48),(40,58),(41,43),(41,44),(41,45),(41,58),(42,49),(42,50),(42,51),(42,52),(42,53),(43,59),(43,61),(44,59),(44,62),(45,59),(45,63),(46,60),(46,61),(47,60),(47,62),(48,60),(48,63),(49,59),(49,60),(50,54),(50,55),(50,59),(51,56),(51,57),(51,59),(52,54),(52,56),(52,60),(53,55),(53,57),(53,60),(54,64),(55,64),(56,64),(57,64),(58,61),(58,62),(58,63),(59,64),(60,64),(61,64),(62,64),(63,64)],65)
=> ? = 3 - 1
[1,5,2,3,4,6] => [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,19),(1,20),(1,21),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(1,46),(2,16),(2,17),(2,18),(2,34),(2,35),(2,36),(2,37),(2,38),(2,39),(2,46),(3,15),(3,18),(3,21),(3,30),(3,33),(3,76),(3,79),(3,80),(4,14),(4,17),(4,20),(4,29),(4,32),(4,75),(4,78),(4,80),(5,13),(5,16),(5,19),(5,28),(5,31),(5,74),(5,77),(5,80),(6,23),(6,26),(6,28),(6,34),(6,40),(6,71),(6,75),(6,76),(7,22),(7,27),(7,29),(7,35),(7,41),(7,72),(7,74),(7,76),(8,24),(8,25),(8,30),(8,36),(8,42),(8,73),(8,74),(8,75),(9,25),(9,27),(9,31),(9,37),(9,43),(9,71),(9,78),(9,79),(10,24),(10,26),(10,32),(10,38),(10,44),(10,72),(10,77),(10,79),(11,22),(11,23),(11,33),(11,39),(11,45),(11,73),(11,77),(11,78),(12,13),(12,14),(12,15),(12,46),(12,71),(12,72),(12,73),(13,81),(13,84),(13,99),(13,127),(14,81),(14,85),(14,100),(14,128),(15,81),(15,86),(15,101),(15,129),(16,47),(16,50),(16,82),(16,84),(16,87),(16,90),(17,48),(17,51),(17,82),(17,85),(17,88),(17,91),(18,49),(18,52),(18,82),(18,86),(18,89),(18,92),(19,53),(19,56),(19,83),(19,84),(19,93),(19,96),(20,54),(20,57),(20,83),(20,85),(20,94),(20,97),(21,55),(21,58),(21,83),(21,86),(21,95),(21,98),(22,59),(22,65),(22,103),(22,107),(22,127),(23,60),(23,66),(23,102),(23,107),(23,128),(24,61),(24,67),(24,104),(24,106),(24,127),(25,62),(25,68),(25,104),(25,105),(25,128),(26,63),(26,69),(26,102),(26,106),(26,129),(27,64),(27,70),(27,103),(27,105),(27,129),(28,47),(28,53),(28,99),(28,102),(28,131),(29,48),(29,54),(29,100),(29,103),(29,131),(30,49),(30,55),(30,101),(30,104),(30,131),(31,50),(31,56),(31,99),(31,105),(31,130),(32,51),(32,57),(32,100),(32,106),(32,130),(33,52),(33,58),(33,101),(33,107),(33,130),(34,47),(34,60),(34,63),(34,88),(34,89),(34,120),(35,48),(35,59),(35,64),(35,87),(35,89),(35,121),(36,49),(36,61),(36,62),(36,87),(36,88),(36,122),(37,50),(37,62),(37,64),(37,91),(37,92),(37,120),(38,51),(38,61),(38,63),(38,90),(38,92),(38,121),(39,52),(39,59),(39,60),(39,90),(39,91),(39,122),(40,53),(40,66),(40,69),(40,94),(40,95),(40,120),(41,54),(41,65),(41,70),(41,93),(41,95),(41,121),(42,55),(42,67),(42,68),(42,93),(42,94),(42,122),(43,56),(43,68),(43,70),(43,97),(43,98),(43,120),(44,57),(44,67),(44,69),(44,96),(44,98),(44,121),(45,58),(45,65),(45,66),(45,96),(45,97),(45,122),(46,84),(46,85),(46,86),(46,120),(46,121),(46,122),(47,108),(47,124),(47,132),(48,109),(48,125),(48,132),(49,110),(49,126),(49,132),(50,111),(50,124),(50,133),(51,112),(51,125),(51,133),(52,113),(52,126),(52,133),(53,114),(53,124),(53,134),(54,115),(54,125),(54,134),(55,116),(55,126),(55,134),(56,117),(56,124),(56,135),(57,118),(57,125),(57,135),(58,119),(58,126),(58,135),(59,109),(59,113),(59,136),(60,108),(60,113),(60,137),(61,110),(61,112),(61,136),(62,110),(62,111),(62,137),(63,108),(63,112),(63,138),(64,109),(64,111),(64,138),(65,115),(65,119),(65,136),(66,114),(66,119),(66,137),(67,116),(67,118),(67,136),(68,116),(68,117),(68,137),(69,114),(69,118),(69,138),(70,115),(70,117),(70,138),(71,99),(71,120),(71,128),(71,129),(72,100),(72,121),(72,127),(72,129),(73,101),(73,122),(73,127),(73,128),(74,87),(74,93),(74,105),(74,127),(74,131),(75,88),(75,94),(75,106),(75,128),(75,131),(76,89),(76,95),(76,107),(76,129),(76,131),(77,90),(77,96),(77,102),(77,127),(77,130),(78,91),(78,97),(78,103),(78,128),(78,130),(79,92),(79,98),(79,104),(79,129),(79,130),(80,81),(80,82),(80,83),(80,130),(80,131),(81,123),(81,139),(82,123),(82,132),(82,133),(83,123),(83,134),(83,135),(84,123),(84,124),(84,136),(85,123),(85,125),(85,137),(86,123),(86,126),(86,138),(87,111),(87,132),(87,136),(88,112),(88,132),(88,137),(89,113),(89,132),(89,138),(90,108),(90,133),(90,136),(91,109),(91,133),(91,137),(92,110),(92,133),(92,138),(93,117),(93,134),(93,136),(94,118),(94,134),(94,137),(95,119),(95,134),(95,138),(96,114),(96,135),(96,136),(97,115),(97,135),(97,137),(98,116),(98,135),(98,138),(99,124),(99,139),(100,125),(100,139),(101,126),(101,139),(102,108),(102,114),(102,139),(103,109),(103,115),(103,139),(104,110),(104,116),(104,139),(105,111),(105,117),(105,139),(106,112),(106,118),(106,139),(107,113),(107,119),(107,139),(108,140),(109,140),(110,140),(111,140),(112,140),(113,140),(114,140),(115,140),(116,140),(117,140),(118,140),(119,140),(120,124),(120,137),(120,138),(121,125),(121,136),(121,138),(122,126),(122,136),(122,137),(123,140),(124,140),(125,140),(126,140),(127,136),(127,139),(128,137),(128,139),(129,138),(129,139),(130,133),(130,135),(130,139),(131,132),(131,134),(131,139),(132,140),(133,140),(134,140),(135,140),(136,140),(137,140),(138,140),(139,140)],141)
=> ? = 3 - 1
[1,5,2,3,6,4] => [4,6,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,16),(1,20),(1,21),(1,32),(1,35),(1,36),(1,41),(1,42),(1,43),(2,15),(2,18),(2,19),(2,31),(2,33),(2,34),(2,39),(2,40),(2,43),(3,23),(3,25),(3,28),(3,38),(3,40),(3,42),(3,71),(3,75),(4,22),(4,24),(4,27),(4,37),(4,39),(4,41),(4,70),(4,75),(5,17),(5,22),(5,23),(5,26),(5,31),(5,32),(5,72),(5,73),(6,13),(6,24),(6,29),(6,33),(6,35),(6,38),(6,72),(6,74),(7,12),(7,25),(7,30),(7,34),(7,36),(7,37),(7,73),(7,74),(8,12),(8,13),(8,14),(8,17),(8,43),(8,70),(8,71),(9,14),(9,15),(9,16),(9,26),(9,44),(9,74),(9,75),(10,18),(10,20),(10,27),(10,30),(10,44),(10,71),(10,72),(11,19),(11,21),(11,28),(11,29),(11,44),(11,70),(11,73),(12,80),(12,88),(12,94),(12,108),(13,80),(13,87),(13,95),(13,109),(14,45),(14,80),(14,83),(14,117),(15,62),(15,81),(15,83),(15,84),(15,92),(16,63),(16,82),(16,83),(16,85),(16,93),(17,45),(17,91),(17,108),(17,109),(18,58),(18,65),(18,84),(18,96),(18,111),(19,59),(19,64),(19,84),(19,97),(19,110),(20,60),(20,67),(20,85),(20,99),(20,111),(21,61),(21,66),(21,85),(21,98),(21,110),(22,46),(22,48),(22,86),(22,89),(22,108),(23,47),(23,49),(23,86),(23,90),(23,109),(24,50),(24,52),(24,87),(24,89),(24,107),(25,51),(25,53),(25,88),(25,90),(25,107),(26,45),(26,62),(26,63),(26,86),(26,118),(27,58),(27,60),(27,68),(27,89),(27,117),(28,59),(28,61),(28,69),(28,90),(28,117),(29,64),(29,66),(29,69),(29,87),(29,118),(30,65),(30,67),(30,68),(30,88),(30,118),(31,46),(31,47),(31,62),(31,91),(31,96),(31,97),(32,48),(32,49),(32,63),(32,91),(32,98),(32,99),(33,50),(33,55),(33,64),(33,92),(33,95),(33,96),(34,51),(34,54),(34,65),(34,92),(34,94),(34,97),(35,52),(35,57),(35,66),(35,93),(35,95),(35,99),(36,53),(36,56),(36,67),(36,93),(36,94),(36,98),(37,54),(37,56),(37,68),(37,107),(37,108),(38,55),(38,57),(38,69),(38,107),(38,109),(39,46),(39,50),(39,54),(39,58),(39,81),(39,110),(40,47),(40,51),(40,55),(40,59),(40,81),(40,111),(41,48),(41,52),(41,56),(41,60),(41,82),(41,110),(42,49),(42,53),(42,57),(42,61),(42,82),(42,111),(43,83),(43,91),(43,94),(43,95),(43,110),(43,111),(44,84),(44,85),(44,117),(44,118),(45,106),(45,124),(46,100),(46,104),(46,121),(47,101),(47,104),(47,122),(48,102),(48,105),(48,121),(49,103),(49,105),(49,122),(50,100),(50,113),(50,115),(51,101),(51,114),(51,115),(52,102),(52,113),(52,116),(53,103),(53,114),(53,116),(54,76),(54,115),(54,121),(55,77),(55,115),(55,122),(56,78),(56,116),(56,121),(57,79),(57,116),(57,122),(58,76),(58,100),(58,123),(59,77),(59,101),(59,123),(60,78),(60,102),(60,123),(61,79),(61,103),(61,123),(62,104),(62,106),(62,119),(63,105),(63,106),(63,120),(64,77),(64,113),(64,119),(65,76),(65,114),(65,119),(66,79),(66,113),(66,120),(67,78),(67,114),(67,120),(68,76),(68,78),(68,124),(69,77),(69,79),(69,124),(70,87),(70,108),(70,110),(70,117),(71,88),(71,109),(71,111),(71,117),(72,89),(72,96),(72,99),(72,109),(72,118),(73,90),(73,97),(73,98),(73,108),(73,118),(74,80),(74,92),(74,93),(74,107),(74,118),(75,81),(75,82),(75,86),(75,107),(75,117),(76,125),(77,125),(78,125),(79,125),(80,112),(80,124),(81,104),(81,115),(81,123),(82,105),(82,116),(82,123),(83,106),(83,112),(83,123),(84,119),(84,123),(85,120),(85,123),(86,104),(86,105),(86,124),(87,113),(87,124),(88,114),(88,124),(89,100),(89,102),(89,124),(90,101),(90,103),(90,124),(91,106),(91,121),(91,122),(92,112),(92,115),(92,119),(93,112),(93,116),(93,120),(94,112),(94,114),(94,121),(95,112),(95,113),(95,122),(96,100),(96,119),(96,122),(97,101),(97,119),(97,121),(98,103),(98,120),(98,121),(99,102),(99,120),(99,122),(100,125),(101,125),(102,125),(103,125),(104,125),(105,125),(106,125),(107,115),(107,116),(107,124),(108,121),(108,124),(109,122),(109,124),(110,113),(110,121),(110,123),(111,114),(111,122),(111,123),(112,125),(113,125),(114,125),(115,125),(116,125),(117,123),(117,124),(118,119),(118,120),(118,124),(119,125),(120,125),(121,125),(122,125),(123,125),(124,125)],126)
=> ? = 3 - 1
[1,5,2,6,3,4] => [4,3,6,2,5,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,11),(1,12),(1,15),(1,58),(1,60),(1,61),(2,15),(2,24),(2,25),(2,26),(2,31),(2,62),(2,63),(3,17),(3,21),(3,28),(3,33),(3,37),(3,61),(3,63),(4,16),(4,20),(4,27),(4,32),(4,37),(4,60),(4,62),(5,18),(5,22),(5,30),(5,34),(5,36),(5,60),(5,63),(6,19),(6,23),(6,29),(6,35),(6,36),(6,61),(6,62),(7,31),(7,32),(7,33),(7,34),(7,35),(7,58),(7,59),(8,13),(8,14),(8,26),(8,27),(8,28),(8,29),(8,30),(8,58),(9,12),(9,14),(9,20),(9,21),(9,22),(9,23),(9,25),(9,59),(10,11),(10,13),(10,16),(10,17),(10,18),(10,19),(10,24),(10,59),(11,40),(11,68),(11,69),(11,94),(12,41),(12,70),(12,71),(12,94),(13,44),(13,45),(13,46),(13,47),(13,56),(13,94),(14,48),(14,49),(14,50),(14,51),(14,57),(14,94),(15,40),(15,41),(15,65),(15,97),(16,44),(16,52),(16,68),(16,72),(16,76),(17,45),(17,52),(17,69),(17,73),(17,77),(18,47),(18,53),(18,68),(18,74),(18,77),(19,46),(19,53),(19,69),(19,75),(19,76),(20,48),(20,54),(20,70),(20,72),(20,78),(21,49),(21,54),(21,71),(21,73),(21,79),(22,50),(22,55),(22,70),(22,74),(22,79),(23,51),(23,55),(23,71),(23,75),(23,78),(24,40),(24,56),(24,64),(24,76),(24,77),(25,41),(25,57),(25,64),(25,78),(25,79),(26,56),(26,57),(26,65),(26,80),(26,81),(27,38),(27,44),(27,48),(27,80),(27,92),(28,38),(28,45),(28,49),(28,81),(28,93),(29,39),(29,46),(29,51),(29,80),(29,93),(30,39),(30,47),(30,50),(30,81),(30,92),(31,64),(31,65),(31,66),(31,67),(32,42),(32,66),(32,72),(32,92),(33,42),(33,67),(33,73),(33,93),(34,43),(34,67),(34,74),(34,92),(35,43),(35,66),(35,75),(35,93),(36,39),(36,43),(36,53),(36,55),(36,97),(37,38),(37,42),(37,52),(37,54),(37,97),(38,82),(38,84),(38,103),(39,83),(39,85),(39,103),(40,98),(40,99),(41,98),(41,100),(42,86),(42,103),(43,87),(43,103),(44,82),(44,88),(44,101),(45,82),(45,89),(45,102),(46,83),(46,88),(46,102),(47,83),(47,89),(47,101),(48,84),(48,90),(48,101),(49,84),(49,91),(49,102),(50,85),(50,91),(50,101),(51,85),(51,90),(51,102),(52,82),(52,86),(52,99),(53,83),(53,87),(53,99),(54,84),(54,86),(54,100),(55,85),(55,87),(55,100),(56,88),(56,89),(56,98),(57,90),(57,91),(57,98),(58,65),(58,92),(58,93),(58,94),(59,64),(59,72),(59,73),(59,74),(59,75),(59,94),(60,68),(60,70),(60,92),(60,97),(61,69),(61,71),(61,93),(61,97),(62,66),(62,76),(62,78),(62,80),(62,97),(63,67),(63,77),(63,79),(63,81),(63,97),(64,95),(64,96),(64,98),(65,98),(65,103),(66,95),(66,103),(67,96),(67,103),(68,99),(68,101),(69,99),(69,102),(70,100),(70,101),(71,100),(71,102),(72,86),(72,95),(72,101),(73,86),(73,96),(73,102),(74,87),(74,96),(74,101),(75,87),(75,95),(75,102),(76,88),(76,95),(76,99),(77,89),(77,96),(77,99),(78,90),(78,95),(78,100),(79,91),(79,96),(79,100),(80,88),(80,90),(80,103),(81,89),(81,91),(81,103),(82,104),(83,104),(84,104),(85,104),(86,104),(87,104),(88,104),(89,104),(90,104),(91,104),(92,101),(92,103),(93,102),(93,103),(94,98),(94,101),(94,102),(95,104),(96,104),(97,99),(97,100),(97,103),(98,104),(99,104),(100,104),(101,104),(102,104),(103,104)],105)
=> ? = 3 - 1
[4,6,5,2,3,1] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[4,6,5,3,1,2] => [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[5,6,3,4,1,2] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[6,3,4,5,2,1] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[6,4,5,2,3,1] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[6,4,5,3,1,2] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[6,5,2,3,4,1] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[6,5,2,4,1,3] => [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[6,5,3,1,4,2] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[6,5,3,4,1,2] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[6,5,4,1,2,3] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,7,6,3,4,2,1] => [1,2,4,3,6,7,5] => ([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[5,7,6,4,2,3,1] => [1,3,2,4,6,7,5] => ([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[5,7,6,4,3,1,2] => [2,1,3,4,6,7,5] => ([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[6,7,4,5,2,3,1] => [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[6,7,4,5,3,1,2] => [2,1,3,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[6,7,5,3,4,1,2] => [2,1,4,3,5,7,6] => ([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[7,4,5,6,3,2,1] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[7,4,6,3,5,2,1] => [1,2,5,3,6,4,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[7,5,3,6,4,2,1] => [1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[7,5,6,3,4,2,1] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[7,5,6,4,2,3,1] => [1,3,2,4,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[7,5,6,4,3,1,2] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[7,6,3,5,2,4,1] => [1,4,2,5,3,6,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[7,6,4,2,5,3,1] => [1,3,5,2,4,6,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[7,6,4,5,2,3,1] => [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[7,6,4,5,3,1,2] => [2,1,3,5,4,6,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[7,6,5,2,3,4,1] => [1,4,3,2,5,6,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[7,6,5,2,4,1,3] => [3,1,4,2,5,6,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[7,6,5,3,1,4,2] => [2,4,1,3,5,6,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 3 - 1
[7,6,5,3,4,1,2] => [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
Description
The cardinality of the Frattini sublattice of a lattice.
The Frattini sublattice is the intersection of all proper maximal sublattices of the lattice.
The following 23 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001820The size of the image of the pop stack sorting operator. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000068The number of minimal elements in a poset. St001626The number of maximal proper sublattices of a lattice. St001568The smallest positive integer that does not appear twice in the partition. St001638The book thickness of a graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000550The number of modular elements of a lattice. St001623The number of doubly irreducible elements of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset.
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