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Matching statistic: St001875
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,1,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,2,5,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,4,3,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,4,5,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,5,1,3,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,5,3,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,5,4,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,4,2,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,4,5,2] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,5,1,2,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,5,2,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,2,5,3] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,3,2,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001812
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 60%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 60%
Values
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,3] => [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[1,2,3,4,5] => [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)
=> 4 = 5 - 1
[1,2,3,5,4] => [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)
=> 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,5,2,3,4] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,5,2,4,3] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3,4,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3,5,1,4] => [4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[2,4,5,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,5,1,3,4] => [3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[2,5,1,4,3] => [3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[2,5,3,1,4] => [4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[2,5,4,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,4,2,5] => [4,2,3,1,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,4,5,2] => [5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,5,2,4] => [4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[3,2,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,4,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,5,1,4] => [4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[3,5,1,2,4] => [4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[3,5,2,1,4] => [4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[4,1,2,3,5] => [4,2,3,1,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,2,5,3] => [5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,3,2,5] => [4,2,3,1,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,4,5,6] => [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)
=> ? = 6 - 1
[1,2,3,4,6,5] => [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)
=> ? = 5 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(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)
=> ? = 5 - 1
[1,2,3,5,6,4] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,4,6,3,5] => [1,2,5,6,3,4] => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => ([(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)
=> ? = 5 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [4,2,6,3,5,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,5,2,4,6] => [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)
=> ? = 5 - 1
[1,3,5,2,6,4] => [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)
=> ? = 4 - 1
[1,3,6,2,4,5] => [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)
=> ? = 4 - 1
[1,3,6,2,5,4] => [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)
=> ? = 4 - 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [4,2,6,3,5,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [4,2,6,3,5,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [6,5,4,3,1,2] => ([(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)
=> ? = 5 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,1,3,5,6,4] => [2,1,3,6,5,4] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,1,3,6,4,5] => [2,1,3,6,5,4] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,1,3,6,5,4] => [2,1,3,6,5,4] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,1,4,6,3,5] => [2,1,5,6,3,4] => [4,3,6,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4 - 1
[2,1,5,3,4,6] => [2,1,5,4,3,6] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,1,5,4,3,6] => [2,1,5,4,3,6] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,3,1,4,5,6] => [3,2,1,4,5,6] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,3,1,4,6,5] => [3,2,1,4,6,5] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,3,1,5,4,6] => [3,2,1,5,4,6] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[2,3,5,1,4,6] => [4,2,5,1,3,6] => [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,3,5,1,6,4] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 3 - 1
[2,3,6,1,4,5] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 3 - 1
[2,3,6,1,5,4] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 3 - 1
Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
Matching statistic: St001960
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 60%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 60%
Values
[1,2,3] => [1,2,3] => [3,2,1] => 1 = 3 - 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 2 = 4 - 2
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1 = 3 - 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 3 - 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1 = 3 - 2
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => 1 = 3 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 3 = 5 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 2 = 4 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 2 = 4 - 2
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => 1 = 3 - 2
[1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => 1 = 3 - 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => 1 = 3 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 2 = 4 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 1 = 3 - 2
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1 = 3 - 2
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,2,4,1] => 1 = 3 - 2
[1,3,5,2,4] => [1,4,5,2,3] => [4,2,5,3,1] => 2 = 4 - 2
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1 = 3 - 2
[1,4,2,5,3] => [1,5,3,4,2] => [3,5,2,4,1] => 1 = 3 - 2
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1 = 3 - 2
[1,4,3,5,2] => [1,5,3,4,2] => [3,5,2,4,1] => 1 = 3 - 2
[1,5,2,3,4] => [1,5,3,4,2] => [3,5,2,4,1] => 1 = 3 - 2
[1,5,2,4,3] => [1,5,3,4,2] => [3,5,2,4,1] => 1 = 3 - 2
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => 2 = 4 - 2
[2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 1 = 3 - 2
[2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 1 = 3 - 2
[2,3,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 1 = 3 - 2
[2,3,4,1,5] => [4,2,3,1,5] => [5,2,4,1,3] => 1 = 3 - 2
[2,3,4,5,1] => [5,2,3,4,1] => [2,5,3,1,4] => 1 = 3 - 2
[2,3,5,1,4] => [4,2,5,1,3] => [2,4,1,5,3] => 1 = 3 - 2
[2,4,1,3,5] => [3,4,1,2,5] => [5,3,1,4,2] => 2 = 4 - 2
[2,4,1,5,3] => [3,5,1,4,2] => [3,1,5,2,4] => 1 = 3 - 2
[2,4,5,1,3] => [4,5,3,1,2] => [4,1,3,5,2] => 1 = 3 - 2
[2,5,1,3,4] => [3,5,1,4,2] => [3,1,5,2,4] => 1 = 3 - 2
[2,5,1,4,3] => [3,5,1,4,2] => [3,1,5,2,4] => 1 = 3 - 2
[2,5,3,1,4] => [4,5,3,1,2] => [4,1,3,5,2] => 1 = 3 - 2
[2,5,4,1,3] => [4,5,3,1,2] => [4,1,3,5,2] => 1 = 3 - 2
[3,1,2,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 1 = 3 - 2
[3,1,4,2,5] => [4,2,3,1,5] => [5,2,4,1,3] => 1 = 3 - 2
[3,1,4,5,2] => [5,2,3,4,1] => [2,5,3,1,4] => 1 = 3 - 2
[3,1,5,2,4] => [4,2,5,1,3] => [2,4,1,5,3] => 1 = 3 - 2
[3,2,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 1 = 3 - 2
[3,2,4,1,5] => [4,2,3,1,5] => [5,2,4,1,3] => 1 = 3 - 2
[3,2,4,5,1] => [5,2,3,4,1] => [2,5,3,1,4] => 1 = 3 - 2
[3,2,5,1,4] => [4,2,5,1,3] => [2,4,1,5,3] => 1 = 3 - 2
[3,5,1,2,4] => [4,5,3,1,2] => [4,1,3,5,2] => 1 = 3 - 2
[3,5,2,1,4] => [4,5,3,1,2] => [4,1,3,5,2] => 1 = 3 - 2
[4,1,2,3,5] => [4,2,3,1,5] => [5,2,4,1,3] => 1 = 3 - 2
[4,1,2,5,3] => [5,2,3,4,1] => [2,5,3,1,4] => 1 = 3 - 2
[4,1,3,2,5] => [4,2,3,1,5] => [5,2,4,1,3] => 1 = 3 - 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ? = 6 - 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => ? = 5 - 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => ? = 5 - 2
[1,2,3,5,6,4] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => ? = 4 - 2
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => ? = 4 - 2
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => ? = 4 - 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => ? = 5 - 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => ? = 4 - 2
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => ? = 4 - 2
[1,2,4,5,6,3] => [1,2,6,4,5,3] => [4,6,3,5,2,1] => ? = 4 - 2
[1,2,4,6,3,5] => [1,2,5,6,3,4] => [5,3,6,4,2,1] => ? = 5 - 2
[1,2,4,6,5,3] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => ? = 3 - 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => ? = 4 - 2
[1,2,5,3,6,4] => [1,2,6,4,5,3] => [4,6,3,5,2,1] => ? = 4 - 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => ? = 4 - 2
[1,2,5,4,6,3] => [1,2,6,4,5,3] => [4,6,3,5,2,1] => ? = 4 - 2
[1,2,5,6,3,4] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => ? = 3 - 2
[1,2,5,6,4,3] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => ? = 3 - 2
[1,2,6,3,4,5] => [1,2,6,4,5,3] => [4,6,3,5,2,1] => ? = 4 - 2
[1,2,6,3,5,4] => [1,2,6,4,5,3] => [4,6,3,5,2,1] => ? = 4 - 2
[1,2,6,4,3,5] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => ? = 3 - 2
[1,2,6,4,5,3] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => ? = 3 - 2
[1,2,6,5,3,4] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => ? = 3 - 2
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => ? = 3 - 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => ? = 5 - 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [5,6,4,2,3,1] => ? = 4 - 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => ? = 4 - 2
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [4,5,6,2,3,1] => ? = 3 - 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [4,5,6,2,3,1] => ? = 3 - 2
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [4,5,6,2,3,1] => ? = 3 - 2
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [6,5,2,3,4,1] => ? = 4 - 2
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [5,6,2,3,4,1] => ? = 3 - 2
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [6,3,5,2,4,1] => ? = 4 - 2
[1,3,4,5,6,2] => [1,6,3,4,5,2] => [3,6,4,2,5,1] => ? = 4 - 2
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [3,5,2,6,4,1] => ? = 4 - 2
[1,3,4,6,5,2] => [1,6,3,5,4,2] => [3,6,2,4,5,1] => ? = 3 - 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [6,4,2,5,3,1] => ? = 5 - 2
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [4,2,6,3,5,1] => ? = 4 - 2
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [6,2,3,4,5,1] => ? = 3 - 2
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [3,4,6,2,5,1] => ? = 3 - 2
[1,3,5,6,2,4] => [1,5,6,4,2,3] => [5,2,4,6,3,1] => ? = 4 - 2
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [4,2,6,3,5,1] => ? = 4 - 2
[1,3,6,2,5,4] => [1,4,6,2,5,3] => [4,2,6,3,5,1] => ? = 4 - 2
[1,3,6,4,2,5] => [1,5,6,4,2,3] => [5,2,4,6,3,1] => ? = 4 - 2
[1,3,6,5,2,4] => [1,5,6,4,2,3] => [5,2,4,6,3,1] => ? = 4 - 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [6,5,2,3,4,1] => ? = 4 - 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [5,6,2,3,4,1] => ? = 3 - 2
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [6,3,5,2,4,1] => ? = 4 - 2
[1,4,2,5,6,3] => [1,6,3,4,5,2] => [3,6,4,2,5,1] => ? = 4 - 2
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [3,5,2,6,4,1] => ? = 4 - 2
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001427
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001427: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 60%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001427: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 60%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => 3
[2,4,1,3] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => 4
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 3
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 3
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 3
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 3
[1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 4
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 3
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 3
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 3
[1,5,2,3,4] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 3
[1,5,2,4,3] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 3
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => 4
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => 3
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => 3
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => 3
[2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => 3
[2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => 3
[2,3,5,1,4] => [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => 3
[2,4,1,3,5] => [3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => 4
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => 3
[2,4,5,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => 3
[2,5,1,3,4] => [3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => 3
[2,5,1,4,3] => [3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => 3
[2,5,3,1,4] => [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => 3
[2,5,4,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => 3
[3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => 3
[3,1,4,2,5] => [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => 3
[3,1,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => 3
[3,1,5,2,4] => [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => 3
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => 3
[3,2,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => 3
[3,2,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => 3
[3,2,5,1,4] => [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => 3
[3,5,1,2,4] => [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => 3
[3,5,2,1,4] => [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => 3
[4,1,2,3,5] => [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => 3
[4,1,2,5,3] => [5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => 3
[4,1,3,2,5] => [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [-1,-2,-3,-4,-5,-6] => ? = 6
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [-1,-2,-3,-4,-6,-5] => ? = 5
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [-1,-2,-3,-5,-4,-6] => ? = 5
[1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [-1,-2,-3,-6,-5,-4] => ? = 4
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [-1,-2,-3,-6,-5,-4] => ? = 4
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [-1,-2,-3,-6,-5,-4] => ? = 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [-1,-2,-4,-3,-5,-6] => ? = 5
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [-1,-2,-4,-3,-6,-5] => ? = 4
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => [-1,-2,-5,-4,-3,-6] => ? = 4
[1,2,4,5,6,3] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => [-1,-2,-6,-4,-5,-3] => ? = 4
[1,2,4,6,3,5] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => [-1,-2,-5,-6,-3,-4] => ? = 5
[1,2,4,6,5,3] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [-1,-2,-6,-5,-4,-3] => ? = 3
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => [-1,-2,-5,-4,-3,-6] => ? = 4
[1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => [-1,-2,-6,-4,-5,-3] => ? = 4
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => [-1,-2,-5,-4,-3,-6] => ? = 4
[1,2,5,4,6,3] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => [-1,-2,-6,-4,-5,-3] => ? = 4
[1,2,5,6,3,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [-1,-2,-6,-5,-4,-3] => ? = 3
[1,2,5,6,4,3] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [-1,-2,-6,-5,-4,-3] => ? = 3
[1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => [-1,-2,-6,-4,-5,-3] => ? = 4
[1,2,6,3,5,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => [-1,-2,-6,-4,-5,-3] => ? = 4
[1,2,6,4,3,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [-1,-2,-6,-5,-4,-3] => ? = 3
[1,2,6,4,5,3] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [-1,-2,-6,-5,-4,-3] => ? = 3
[1,2,6,5,3,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [-1,-2,-6,-5,-4,-3] => ? = 3
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [-1,-2,-6,-5,-4,-3] => ? = 3
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [-1,-3,-2,-4,-5,-6] => ? = 5
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [-1,-3,-2,-4,-6,-5] => ? = 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [-1,-3,-2,-5,-4,-6] => ? = 4
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => [-1,-3,-2,-6,-5,-4] => ? = 3
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => [-1,-3,-2,-6,-5,-4] => ? = 3
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => [-1,-3,-2,-6,-5,-4] => ? = 3
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => [-1,-4,-3,-2,-5,-6] => ? = 4
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => [-1,-4,-3,-2,-6,-5] => ? = 3
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => [-1,-5,-3,-4,-2,-6] => ? = 4
[1,3,4,5,6,2] => [1,6,3,4,5,2] => [1,6,3,4,5,2] => [-1,-6,-3,-4,-5,-2] => ? = 4
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,5,3,6,2,4] => ? => ? = 4
[1,3,4,6,5,2] => [1,6,3,5,4,2] => [1,6,3,5,4,2] => [-1,-6,-3,-5,-4,-2] => ? = 3
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => [-1,-4,-5,-2,-3,-6] => ? = 5
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? => ? = 4
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [-1,-5,-4,-3,-2,-6] => ? = 3
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [1,6,4,3,5,2] => [-1,-6,-4,-3,-5,-2] => ? = 3
[1,3,5,6,2,4] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => ? => ? = 4
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? => ? = 4
[1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? => ? = 4
[1,3,6,4,2,5] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => ? => ? = 4
[1,3,6,5,2,4] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => ? => ? = 4
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => [-1,-4,-3,-2,-5,-6] => ? = 4
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => [-1,-4,-3,-2,-6,-5] => ? = 3
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => [-1,-5,-3,-4,-2,-6] => ? = 4
[1,4,2,5,6,3] => [1,6,3,4,5,2] => [1,6,3,4,5,2] => [-1,-6,-3,-4,-5,-2] => ? = 4
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,5,3,6,2,4] => ? => ? = 4
Description
The number of descents of a signed permutation.
A descent of a signed permutation $\sigma$ of length $n$ is an index $0 \leq i < n$ such that $\sigma(i) > \sigma(i+1)$, setting $\sigma(0) = 0$.
Matching statistic: St001645
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => ([],3)
=> ? = 3 + 4
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => ([],4)
=> ? = 4 + 4
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 3 + 4
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 3 + 4
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 3 + 4
[2,4,1,3] => {{1,2,4},{3}}
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 + 4
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([],5)
=> ? = 5 + 4
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 4 + 4
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? = 4 + 4
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 4
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? = 4 + 4
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 3 + 4
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 3 + 4
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 4
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 4
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[1,5,2,4,3] => {{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? = 4 + 4
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 3 + 4
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 3 + 4
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 3 + 4
[2,3,4,1,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 4
[2,3,4,5,1] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 3 + 4
[2,3,5,1,4] => {{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[2,4,1,3,5] => {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 4
[2,4,1,5,3] => {{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[2,4,5,1,3] => {{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[2,5,1,3,4] => {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[2,5,1,4,3] => {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[2,5,3,1,4] => {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[2,5,4,1,3] => {{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,1,4,2,5] => {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,1,4,5,2] => {{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,1,5,2,4] => {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,2,4,1,5] => {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,2,4,5,1] => {{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,2,5,1,4] => {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,5,1,2,4] => {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[3,5,2,1,4] => {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[4,1,2,5,3] => {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 4
[4,6,2,7,1,3,5] => {{1,4,7},{2,6},{3},{5}}
=> [4,6,3,7,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[4,6,2,7,3,1,5] => {{1,4,7},{2,6},{3},{5}}
=> [4,6,3,7,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[4,6,3,7,1,2,5] => {{1,4,7},{2,6},{3},{5}}
=> [4,6,3,7,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[4,6,3,7,2,1,5] => {{1,4,7},{2,6},{3},{5}}
=> [4,6,3,7,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[4,6,7,1,2,3,5] => {{1,4},{2,6},{3,7},{5}}
=> [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,6,7,1,3,2,5] => {{1,4},{2,6},{3,7},{5}}
=> [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,6,7,2,1,3,5] => {{1,4},{2,6},{3,7},{5}}
=> [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,6,7,2,3,1,5] => {{1,4},{2,6},{3,7},{5}}
=> [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,6,7,3,1,2,5] => {{1,4},{2,6},{3,7},{5}}
=> [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,6,7,3,2,1,5] => {{1,4},{2,6},{3,7},{5}}
=> [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,7,2,6,1,3,5] => {{1,4,6},{2,7},{3},{5}}
=> [4,7,3,6,5,1,2] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,7,2,6,3,1,5] => {{1,4,6},{2,7},{3},{5}}
=> [4,7,3,6,5,1,2] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,7,3,6,1,2,5] => {{1,4,6},{2,7},{3},{5}}
=> [4,7,3,6,5,1,2] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,7,3,6,2,1,5] => {{1,4,6},{2,7},{3},{5}}
=> [4,7,3,6,5,1,2] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[4,7,6,1,2,3,5] => {{1,4},{2,7},{3,6},{5}}
=> [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[4,7,6,1,3,2,5] => {{1,4},{2,7},{3,6},{5}}
=> [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[4,7,6,2,1,3,5] => {{1,4},{2,7},{3,6},{5}}
=> [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[4,7,6,2,3,1,5] => {{1,4},{2,7},{3,6},{5}}
=> [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[4,7,6,3,1,2,5] => {{1,4},{2,7},{3,6},{5}}
=> [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[4,7,6,3,2,1,5] => {{1,4},{2,7},{3,6},{5}}
=> [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[5,7,1,2,6,3,4] => {{1,5,6},{2,7},{3},{4}}
=> [5,7,3,4,6,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 3 + 4
[5,7,1,2,6,4,3] => {{1,5,6},{2,7},{3},{4}}
=> [5,7,3,4,6,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 3 + 4
[6,3,7,1,2,4,5] => {{1,6},{2,3,7},{4},{5}}
=> [6,3,7,4,5,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 3 + 4
[6,3,7,1,2,5,4] => {{1,6},{2,3,7},{4},{5}}
=> [6,3,7,4,5,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 3 + 4
[6,3,7,2,1,4,5] => {{1,6},{2,3,7},{4},{5}}
=> [6,3,7,4,5,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 3 + 4
[6,3,7,2,1,5,4] => {{1,6},{2,3,7},{4},{5}}
=> [6,3,7,4,5,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 3 + 4
[6,7,1,2,3,4,5] => {{1,6},{2,7},{3},{4},{5}}
=> [6,7,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[6,7,1,2,3,5,4] => {{1,6},{2,7},{3},{4},{5}}
=> [6,7,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[6,7,1,2,4,3,5] => {{1,6},{2,7},{3},{4},{5}}
=> [6,7,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[6,7,1,2,4,5,3] => {{1,6},{2,7},{3},{4},{5}}
=> [6,7,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[6,7,1,2,5,3,4] => {{1,6},{2,7},{3},{4},{5}}
=> [6,7,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[6,7,1,2,5,4,3] => {{1,6},{2,7},{3},{4},{5}}
=> [6,7,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 7 = 3 + 4
[7,3,6,1,2,4,5] => {{1,7},{2,3,6},{4},{5}}
=> [7,3,6,4,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,3,6,1,2,5,4] => {{1,7},{2,3,6},{4},{5}}
=> [7,3,6,4,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,3,6,2,1,4,5] => {{1,7},{2,3,6},{4},{5}}
=> [7,3,6,4,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,3,6,2,1,5,4] => {{1,7},{2,3,6},{4},{5}}
=> [7,3,6,4,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,5,1,2,6,3,4] => {{1,7},{2,5,6},{3},{4}}
=> [7,5,3,4,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,5,1,2,6,4,3] => {{1,7},{2,5,6},{3},{4}}
=> [7,5,3,4,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,6,1,2,3,4,5] => {{1,7},{2,6},{3},{4},{5}}
=> [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,6,1,2,3,5,4] => {{1,7},{2,6},{3},{4},{5}}
=> [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,6,1,2,4,3,5] => {{1,7},{2,6},{3},{4},{5}}
=> [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,6,1,2,4,5,3] => {{1,7},{2,6},{3},{4},{5}}
=> [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,6,1,2,5,3,4] => {{1,7},{2,6},{3},{4},{5}}
=> [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
[7,6,1,2,5,4,3] => {{1,7},{2,6},{3},{4},{5}}
=> [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 3 + 4
Description
The pebbling number of a connected graph.
Matching statistic: St001616
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00159: Permutations —Demazure product with inverse⟶ 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] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ([(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)
=> ? = 4 - 1
[1,2,4,3] => [3,4,2,1] => [4,3,2,1] => ([(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
[1,3,2,4] => [4,2,3,1] => [4,3,2,1] => ([(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
[2,1,3,4] => [4,3,1,2] => [4,3,2,1] => ([(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
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 3 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 5 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,3,2,1] => ([(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
[1,3,2,5,4] => [4,5,2,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => ([(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
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,5,2,3,4] => [4,3,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [3,4,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,2,1] => ([(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
[2,1,3,5,4] => [4,5,3,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[2,1,4,3,5] => [5,3,4,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[2,3,4,1,5] => [5,1,4,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 1
[2,3,5,1,4] => [4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,4,1,3,5] => [5,3,1,4,2] => [5,4,3,2,1] => ([(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
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[2,4,5,1,3] => [3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,5,1,3,4] => [4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,5,1,4,3] => [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,5,3,1,4] => [4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[2,5,4,1,3] => [3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[3,1,4,2,5] => [5,2,4,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[3,1,4,5,2] => [2,5,4,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 1
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[3,2,4,1,5] => [5,1,4,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 1
[3,2,5,1,4] => [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,5,1,2,4] => [4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,5,2,1,4] => [4,1,2,5,3] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[4,1,2,3,5] => [5,3,2,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[4,1,2,5,3] => [3,5,2,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 1
[4,1,3,2,5] => [5,2,3,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[4,1,3,5,2] => [2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 1
[5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => ([(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)
=> ? = 3 - 1
[3,4,6,1,5,2] => [2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(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
[4,3,6,1,5,2] => [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(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
[5,2,6,1,3,4] => [4,3,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[5,2,6,1,4,3] => [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[4,5,1,6,2,7,3] => [3,7,2,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,6,3,7,2] => [2,7,3,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,2,6,3] => [3,6,2,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,3,6,2] => [2,6,3,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,6,1,7,3] => [3,7,1,6,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,7,1,6,3] => [3,6,1,7,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,2,7,3] => [3,7,2,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,3,7,2] => [2,7,3,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[5,1,6,2,7,3,4] => [4,3,7,2,6,1,5] => [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
[5,1,6,2,7,4,3] => [3,4,7,2,6,1,5] => [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
[5,1,6,3,7,2,4] => [4,2,7,3,6,1,5] => [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
[5,1,6,3,7,4,2] => [2,4,7,3,6,1,5] => [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
[5,1,7,2,6,3,4] => [4,3,6,2,7,1,5] => [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
[5,1,7,2,6,4,3] => [3,4,6,2,7,1,5] => [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
[5,2,6,1,7,3,4] => [4,3,7,1,6,2,5] => [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
[5,2,6,1,7,4,3] => [3,4,7,1,6,2,5] => [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
[5,2,7,1,6,3,4] => [4,3,6,1,7,2,5] => [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
[5,2,7,1,6,4,3] => [3,4,6,1,7,2,5] => [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
[5,4,1,6,2,7,3] => [3,7,2,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,6,3,7,2] => [2,7,3,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,2,6,3] => [3,6,2,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,3,6,2] => [2,6,3,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,6,1,7,3] => [3,7,1,6,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,7,1,6,3] => [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,6,1,4,7,2,3] => [3,2,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[5,6,1,4,7,3,2] => [2,3,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[6,1,5,2,7,3,4] => [4,3,7,2,5,1,6] => [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
[6,1,5,2,7,4,3] => [3,4,7,2,5,1,6] => [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
[6,1,5,3,7,2,4] => [4,2,7,3,5,1,6] => [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
[6,1,5,3,7,4,2] => [2,4,7,3,5,1,6] => [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
[6,2,5,1,7,3,4] => [4,3,7,1,5,2,6] => [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
[6,2,5,1,7,4,3] => [3,4,7,1,5,2,6] => [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
[6,4,1,5,2,7,3] => [3,7,2,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,4,1,5,3,7,2] => [2,7,3,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,5,1,4,7,2,3] => [3,2,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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
[6,5,1,4,7,3,2] => [2,3,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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: St001720
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ([(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)
=> ? = 4 - 1
[1,2,4,3] => [3,4,2,1] => [4,3,2,1] => ([(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
[1,3,2,4] => [4,2,3,1] => [4,3,2,1] => ([(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
[2,1,3,4] => [4,3,1,2] => [4,3,2,1] => ([(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
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 3 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 5 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,3,2,1] => ([(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
[1,3,2,5,4] => [4,5,2,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => ([(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
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[1,5,2,3,4] => [4,3,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [3,4,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,2,1] => ([(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
[2,1,3,5,4] => [4,5,3,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[2,1,4,3,5] => [5,3,4,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[2,3,4,1,5] => [5,1,4,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 1
[2,3,5,1,4] => [4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,4,1,3,5] => [5,3,1,4,2] => [5,4,3,2,1] => ([(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
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[2,4,5,1,3] => [3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,5,1,3,4] => [4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,5,1,4,3] => [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[2,5,3,1,4] => [4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[2,5,4,1,3] => [3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[3,1,4,2,5] => [5,2,4,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[3,1,4,5,2] => [2,5,4,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 1
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[3,2,4,1,5] => [5,1,4,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 1
[3,2,5,1,4] => [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,5,1,2,4] => [4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,5,2,1,4] => [4,1,2,5,3] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[4,1,2,3,5] => [5,3,2,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[4,1,2,5,3] => [3,5,2,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 1
[4,1,3,2,5] => [5,2,3,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 1
[4,1,3,5,2] => [2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 1
[5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => ([(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)
=> ? = 3 - 1
[3,4,6,1,5,2] => [2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(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
[4,3,6,1,5,2] => [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(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
[5,2,6,1,3,4] => [4,3,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[5,2,6,1,4,3] => [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[4,5,1,6,2,7,3] => [3,7,2,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,6,3,7,2] => [2,7,3,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,2,6,3] => [3,6,2,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,3,6,2] => [2,6,3,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,6,1,7,3] => [3,7,1,6,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,7,1,6,3] => [3,6,1,7,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,2,7,3] => [3,7,2,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,3,7,2] => [2,7,3,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[5,1,6,2,7,3,4] => [4,3,7,2,6,1,5] => [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
[5,1,6,2,7,4,3] => [3,4,7,2,6,1,5] => [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
[5,1,6,3,7,2,4] => [4,2,7,3,6,1,5] => [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
[5,1,6,3,7,4,2] => [2,4,7,3,6,1,5] => [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
[5,1,7,2,6,3,4] => [4,3,6,2,7,1,5] => [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
[5,1,7,2,6,4,3] => [3,4,6,2,7,1,5] => [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
[5,2,6,1,7,3,4] => [4,3,7,1,6,2,5] => [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
[5,2,6,1,7,4,3] => [3,4,7,1,6,2,5] => [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
[5,2,7,1,6,3,4] => [4,3,6,1,7,2,5] => [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
[5,2,7,1,6,4,3] => [3,4,6,1,7,2,5] => [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
[5,4,1,6,2,7,3] => [3,7,2,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,6,3,7,2] => [2,7,3,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,2,6,3] => [3,6,2,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,3,6,2] => [2,6,3,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,6,1,7,3] => [3,7,1,6,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,7,1,6,3] => [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,6,1,4,7,2,3] => [3,2,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[5,6,1,4,7,3,2] => [2,3,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[6,1,5,2,7,3,4] => [4,3,7,2,5,1,6] => [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
[6,1,5,2,7,4,3] => [3,4,7,2,5,1,6] => [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
[6,1,5,3,7,2,4] => [4,2,7,3,5,1,6] => [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
[6,1,5,3,7,4,2] => [2,4,7,3,5,1,6] => [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
[6,2,5,1,7,3,4] => [4,3,7,1,5,2,6] => [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
[6,2,5,1,7,4,3] => [3,4,7,1,5,2,6] => [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
[6,4,1,5,2,7,3] => [3,7,2,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,4,1,5,3,7,2] => [2,7,3,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,5,1,4,7,2,3] => [3,2,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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
[6,5,1,4,7,3,2] => [2,3,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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 minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001613
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00159: Permutations —Demazure product with inverse⟶ 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] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ([(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)
=> ? = 4 - 2
[1,2,4,3] => [3,4,2,1] => [4,3,2,1] => ([(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
[1,3,2,4] => [4,2,3,1] => [4,3,2,1] => ([(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
[2,1,3,4] => [4,3,1,2] => [4,3,2,1] => ([(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
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 3 - 2
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 5 - 2
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,3,2,1] => ([(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
[1,3,2,5,4] => [4,5,2,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,4,5,2] => [2,5,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => ([(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
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,4,3,5,2] => [2,5,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,5,2,3,4] => [4,3,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[1,5,2,4,3] => [3,4,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,2,1] => ([(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
[2,1,3,5,4] => [4,5,3,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,1,4,3,5] => [5,3,4,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,3,4,1,5] => [5,1,4,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 2
[2,3,5,1,4] => [4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,4,1,3,5] => [5,3,1,4,2] => [5,4,3,2,1] => ([(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
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,4,5,1,3] => [3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,5,1,3,4] => [4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,5,1,4,3] => [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,5,3,1,4] => [4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 2
[2,5,4,1,3] => [3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 2
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,1,4,2,5] => [5,2,4,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,1,4,5,2] => [2,5,4,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 2
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,2,4,1,5] => [5,1,4,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 2
[3,2,5,1,4] => [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[3,5,1,2,4] => [4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[3,5,2,1,4] => [4,1,2,5,3] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 2
[4,1,2,3,5] => [5,3,2,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[4,1,2,5,3] => [3,5,2,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 2
[4,1,3,2,5] => [5,2,3,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[4,1,3,5,2] => [2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 2
[5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => ([(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)
=> ? = 3 - 2
[3,4,6,1,5,2] => [2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(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
[4,3,6,1,5,2] => [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(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
[5,2,6,1,3,4] => [4,3,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[5,2,6,1,4,3] => [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[4,5,1,6,2,7,3] => [3,7,2,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,6,3,7,2] => [2,7,3,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,2,6,3] => [3,6,2,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,3,6,2] => [2,6,3,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,6,1,7,3] => [3,7,1,6,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,7,1,6,3] => [3,6,1,7,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,2,7,3] => [3,7,2,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,3,7,2] => [2,7,3,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[5,1,6,2,7,3,4] => [4,3,7,2,6,1,5] => [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
[5,1,6,2,7,4,3] => [3,4,7,2,6,1,5] => [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
[5,1,6,3,7,2,4] => [4,2,7,3,6,1,5] => [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
[5,1,6,3,7,4,2] => [2,4,7,3,6,1,5] => [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
[5,1,7,2,6,3,4] => [4,3,6,2,7,1,5] => [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
[5,1,7,2,6,4,3] => [3,4,6,2,7,1,5] => [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
[5,2,6,1,7,3,4] => [4,3,7,1,6,2,5] => [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
[5,2,6,1,7,4,3] => [3,4,7,1,6,2,5] => [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
[5,2,7,1,6,3,4] => [4,3,6,1,7,2,5] => [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
[5,2,7,1,6,4,3] => [3,4,6,1,7,2,5] => [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
[5,4,1,6,2,7,3] => [3,7,2,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,6,3,7,2] => [2,7,3,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,2,6,3] => [3,6,2,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,3,6,2] => [2,6,3,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,6,1,7,3] => [3,7,1,6,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,7,1,6,3] => [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,6,1,4,7,2,3] => [3,2,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[5,6,1,4,7,3,2] => [2,3,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[6,1,5,2,7,3,4] => [4,3,7,2,5,1,6] => [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
[6,1,5,2,7,4,3] => [3,4,7,2,5,1,6] => [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
[6,1,5,3,7,2,4] => [4,2,7,3,5,1,6] => [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
[6,1,5,3,7,4,2] => [2,4,7,3,5,1,6] => [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
[6,2,5,1,7,3,4] => [4,3,7,1,5,2,6] => [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
[6,2,5,1,7,4,3] => [3,4,7,1,5,2,6] => [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
[6,4,1,5,2,7,3] => [3,7,2,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,4,1,5,3,7,2] => [2,7,3,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,5,1,4,7,2,3] => [3,2,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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
[6,5,1,4,7,3,2] => [2,3,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00159: Permutations —Demazure product with inverse⟶ 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] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ([(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)
=> ? = 4 - 2
[1,2,4,3] => [3,4,2,1] => [4,3,2,1] => ([(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
[1,3,2,4] => [4,2,3,1] => [4,3,2,1] => ([(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
[2,1,3,4] => [4,3,1,2] => [4,3,2,1] => ([(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
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 3 - 2
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 5 - 2
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,3,2,1] => ([(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
[1,3,2,5,4] => [4,5,2,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,4,5,2] => [2,5,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => ([(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
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,4,3,5,2] => [2,5,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,5,2,3,4] => [4,3,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[1,5,2,4,3] => [3,4,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,2,1] => ([(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
[2,1,3,5,4] => [4,5,3,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,1,4,3,5] => [5,3,4,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,3,4,1,5] => [5,1,4,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 2
[2,3,5,1,4] => [4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,4,1,3,5] => [5,3,1,4,2] => [5,4,3,2,1] => ([(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
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,4,5,1,3] => [3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,5,1,3,4] => [4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,5,1,4,3] => [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,5,3,1,4] => [4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 2
[2,5,4,1,3] => [3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 2
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,1,4,2,5] => [5,2,4,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,1,4,5,2] => [2,5,4,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 2
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,2,4,1,5] => [5,1,4,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 2
[3,2,5,1,4] => [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[3,5,1,2,4] => [4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[3,5,2,1,4] => [4,1,2,5,3] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 2
[4,1,2,3,5] => [5,3,2,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[4,1,2,5,3] => [3,5,2,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 2
[4,1,3,2,5] => [5,2,3,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[4,1,3,5,2] => [2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 2
[5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => ([(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)
=> ? = 3 - 2
[3,4,6,1,5,2] => [2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(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
[4,3,6,1,5,2] => [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(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
[5,2,6,1,3,4] => [4,3,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[5,2,6,1,4,3] => [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[4,5,1,6,2,7,3] => [3,7,2,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,6,3,7,2] => [2,7,3,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,2,6,3] => [3,6,2,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,3,6,2] => [2,6,3,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,6,1,7,3] => [3,7,1,6,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,7,1,6,3] => [3,6,1,7,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,2,7,3] => [3,7,2,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,3,7,2] => [2,7,3,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[5,1,6,2,7,3,4] => [4,3,7,2,6,1,5] => [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
[5,1,6,2,7,4,3] => [3,4,7,2,6,1,5] => [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
[5,1,6,3,7,2,4] => [4,2,7,3,6,1,5] => [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
[5,1,6,3,7,4,2] => [2,4,7,3,6,1,5] => [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
[5,1,7,2,6,3,4] => [4,3,6,2,7,1,5] => [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
[5,1,7,2,6,4,3] => [3,4,6,2,7,1,5] => [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
[5,2,6,1,7,3,4] => [4,3,7,1,6,2,5] => [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
[5,2,6,1,7,4,3] => [3,4,7,1,6,2,5] => [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
[5,2,7,1,6,3,4] => [4,3,6,1,7,2,5] => [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
[5,2,7,1,6,4,3] => [3,4,6,1,7,2,5] => [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
[5,4,1,6,2,7,3] => [3,7,2,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,6,3,7,2] => [2,7,3,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,2,6,3] => [3,6,2,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,3,6,2] => [2,6,3,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,6,1,7,3] => [3,7,1,6,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,7,1,6,3] => [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,6,1,4,7,2,3] => [3,2,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[5,6,1,4,7,3,2] => [2,3,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[6,1,5,2,7,3,4] => [4,3,7,2,5,1,6] => [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
[6,1,5,2,7,4,3] => [3,4,7,2,5,1,6] => [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
[6,1,5,3,7,2,4] => [4,2,7,3,5,1,6] => [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
[6,1,5,3,7,4,2] => [2,4,7,3,5,1,6] => [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
[6,2,5,1,7,3,4] => [4,3,7,1,5,2,6] => [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
[6,2,5,1,7,4,3] => [3,4,7,1,5,2,6] => [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
[6,4,1,5,2,7,3] => [3,7,2,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,4,1,5,3,7,2] => [2,7,3,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,5,1,4,7,2,3] => [3,2,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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
[6,5,1,4,7,3,2] => [2,3,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00159: Permutations —Demazure product with inverse⟶ 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] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ([(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)
=> ? = 4 - 2
[1,2,4,3] => [3,4,2,1] => [4,3,2,1] => ([(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
[1,3,2,4] => [4,2,3,1] => [4,3,2,1] => ([(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
[2,1,3,4] => [4,3,1,2] => [4,3,2,1] => ([(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
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 3 - 2
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 5 - 2
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,3,2,1] => ([(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
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,3,2,1] => ([(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
[1,3,2,5,4] => [4,5,2,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,4,5,2] => [2,5,4,3,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => ([(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
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,4,3,5,2] => [2,5,3,4,1] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[1,5,2,3,4] => [4,3,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[1,5,2,4,3] => [3,4,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,2,1] => ([(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
[2,1,3,5,4] => [4,5,3,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,1,4,3,5] => [5,3,4,1,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,3,4,1,5] => [5,1,4,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 2
[2,3,5,1,4] => [4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,4,1,3,5] => [5,3,1,4,2] => [5,4,3,2,1] => ([(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
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[2,4,5,1,3] => [3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,5,1,3,4] => [4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,5,1,4,3] => [3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[2,5,3,1,4] => [4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 2
[2,5,4,1,3] => [3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 2
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,1,4,2,5] => [5,2,4,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,1,4,5,2] => [2,5,4,1,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 2
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[3,2,4,1,5] => [5,1,4,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => ([(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)
=> ? = 3 - 2
[3,2,5,1,4] => [4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[3,5,1,2,4] => [4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 2
[3,5,2,1,4] => [4,1,2,5,3] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 2
[4,1,2,3,5] => [5,3,2,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[4,1,2,5,3] => [3,5,2,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 2
[4,1,3,2,5] => [5,2,3,1,4] => [5,4,3,2,1] => ([(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)
=> ? = 3 - 2
[4,1,3,5,2] => [2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 2
[5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => ([(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)
=> ? = 3 - 2
[3,4,6,1,5,2] => [2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(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
[4,3,6,1,5,2] => [2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(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
[5,2,6,1,3,4] => [4,3,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[5,2,6,1,4,3] => [3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(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
[4,5,1,6,2,7,3] => [3,7,2,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,6,3,7,2] => [2,7,3,6,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,2,6,3] => [3,6,2,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,1,7,3,6,2] => [2,6,3,7,1,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,6,1,7,3] => [3,7,1,6,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,5,2,7,1,6,3] => [3,6,1,7,2,5,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,2,7,3] => [3,7,2,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[4,6,1,5,3,7,2] => [2,7,3,5,1,6,4] => [5,7,3,6,1,4,2] => ([(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
[5,1,6,2,7,3,4] => [4,3,7,2,6,1,5] => [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
[5,1,6,2,7,4,3] => [3,4,7,2,6,1,5] => [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
[5,1,6,3,7,2,4] => [4,2,7,3,6,1,5] => [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
[5,1,6,3,7,4,2] => [2,4,7,3,6,1,5] => [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
[5,1,7,2,6,3,4] => [4,3,6,2,7,1,5] => [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
[5,1,7,2,6,4,3] => [3,4,6,2,7,1,5] => [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
[5,2,6,1,7,3,4] => [4,3,7,1,6,2,5] => [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
[5,2,6,1,7,4,3] => [3,4,7,1,6,2,5] => [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
[5,2,7,1,6,3,4] => [4,3,6,1,7,2,5] => [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
[5,2,7,1,6,4,3] => [3,4,6,1,7,2,5] => [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
[5,4,1,6,2,7,3] => [3,7,2,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,6,3,7,2] => [2,7,3,6,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,2,6,3] => [3,6,2,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,1,7,3,6,2] => [2,6,3,7,1,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,6,1,7,3] => [3,7,1,6,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,4,2,7,1,6,3] => [3,6,1,7,2,4,5] => [5,7,3,6,1,4,2] => ([(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
[5,6,1,4,7,2,3] => [3,2,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[5,6,1,4,7,3,2] => [2,3,7,4,1,6,5] => [5,2,7,4,1,6,3] => ([(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
[6,1,5,2,7,3,4] => [4,3,7,2,5,1,6] => [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
[6,1,5,2,7,4,3] => [3,4,7,2,5,1,6] => [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
[6,1,5,3,7,2,4] => [4,2,7,3,5,1,6] => [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
[6,1,5,3,7,4,2] => [2,4,7,3,5,1,6] => [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
[6,2,5,1,7,3,4] => [4,3,7,1,5,2,6] => [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
[6,2,5,1,7,4,3] => [3,4,7,1,5,2,6] => [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
[6,4,1,5,2,7,3] => [3,7,2,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,4,1,5,3,7,2] => [2,7,3,5,1,4,6] => [5,7,3,6,1,4,2] => ([(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
[6,5,1,4,7,2,3] => [3,2,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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
[6,5,1,4,7,3,2] => [2,3,7,4,1,5,6] => [5,2,7,4,1,6,3] => ([(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.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St000181The number of connected components of the Hasse diagram for the poset. St000759The smallest missing part in an integer partition. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001863The number of weak excedances of a signed permutation. St001896The number of right descents of a signed permutations.
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