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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000422
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Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 4
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 4
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 4
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 4
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 4
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 4
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 4
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 2
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 4
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 4
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 6
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 6
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000830
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000830: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 86%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000830: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [3,2,1] => [3,1,2] => [3,1,2] => 4
[3,1,2] => [3,2,1] => [3,1,2] => [3,1,2] => 4
[3,2,1] => [3,2,1] => [3,1,2] => [3,1,2] => 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 4
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 4
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 4
[2,4,3,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6
[3,1,2,4] => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 4
[3,2,1,4] => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 4
[3,4,1,2] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6
[3,4,2,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6
[4,2,1,3] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6
[4,2,3,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6
[4,3,1,2] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 4
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 4
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 6
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 4
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 6
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 6
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 6
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 6
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 6
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 6
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 4
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 4
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 6
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 6
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 6
[1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 6
[1,3,5,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 8
[1,3,5,6,4,2,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,3,5,7,6,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,3,6,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,3,6,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,3,6,7,4,5,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,3,7,5,4,6,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,3,7,5,6,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,3,7,6,4,5,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,3,7,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 6
[1,4,5,2,3,7,6] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 8
[1,4,5,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 6
[1,4,5,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 8
[1,4,5,6,2,3,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,4,5,6,3,2,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,4,5,7,6,2,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,5,7,6,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,6,2,5,3,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,4,6,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,4,6,5,2,3,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,4,6,5,3,2,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,4,6,7,2,5,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,6,7,3,5,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,6,7,5,2,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,6,7,5,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,7,5,2,6,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,7,5,3,6,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,7,5,6,2,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,7,5,6,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,7,6,2,5,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,7,6,3,5,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,7,6,5,2,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,4,7,6,5,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,5,3,2,4,6,7] => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 6
[1,5,3,2,4,7,6] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 8
[1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 6
[1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 8
[1,5,3,6,2,4,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,5,3,6,4,2,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
[1,5,3,7,6,2,4] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,5,3,7,6,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 10
[1,5,4,2,3,6,7] => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 6
[1,5,4,2,3,7,6] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 8
[1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 6
[1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 8
[1,5,4,6,2,3,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 8
Description
The total displacement of a permutation.
This is, for a permutation $\pi$ of $n$, given by $\sum_{i = 1}^n | \pi(i) - i |.$
This is twice the statistic [[St000029]] and can be found in [3, Problem 5.1.1.28] and also in [1, 2].
Matching statistic: St000824
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000824: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 86%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000824: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 4
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 4
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 4
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 4
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 4
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 4
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 4
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 4
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 4
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 4
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 4
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 6
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 6
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 6
[1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 6
[1,3,5,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 8
[1,3,5,6,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,3,5,7,2,4,6] => [1,5,6,7,2,3,4] => [1,3,4,7,2,5,6] => [1,5,3,7,2,6,4] => ? = 6
[1,3,5,7,6,4,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,3,6,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,3,6,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,3,6,7,4,5,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,3,7,5,4,6,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,3,7,5,6,4,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,3,7,6,4,5,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,3,7,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 6
[1,4,5,2,3,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 8
[1,4,5,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 6
[1,4,5,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 8
[1,4,5,6,2,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,4,5,6,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,4,5,7,6,2,3] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,5,7,6,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,6,2,5,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,4,6,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,4,6,5,2,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,4,6,5,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,4,6,7,2,5,3] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,6,7,3,5,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,6,7,5,2,3] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,6,7,5,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,7,5,2,6,3] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,7,5,3,6,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,7,5,6,2,3] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,7,5,6,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,7,6,2,5,3] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,7,6,3,5,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,7,6,5,2,3] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,4,7,6,5,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,5,3,2,4,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 6
[1,5,3,2,4,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 8
[1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 6
[1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 8
[1,5,3,6,2,4,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,5,3,6,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 8
[1,5,3,7,6,2,4] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,5,3,7,6,4,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 10
[1,5,4,2,3,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 6
[1,5,4,2,3,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 8
[1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 6
[1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 8
Description
The sum of the number of descents and the number of recoils of a permutation.
This statistic is the sum of [[St000021]] and [[St000354]].
Matching statistic: St001630
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ? = 0 - 2
[1,2] => ([],2)
=> ([],1)
=> ? = 0 - 2
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 2
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 2
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 6 - 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 6 - 2
[1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 6 - 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 6 - 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 8 - 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 8 - 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 8 - 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 2
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 6 - 2
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8 - 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 8 - 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 8 - 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8 - 2
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 8 - 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 8 - 2
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 8 - 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ? = 0 - 2
[1,2] => ([],2)
=> ([],1)
=> ? = 0 - 2
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 2
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 2
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 6 - 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 6 - 2
[1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 6 - 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 6 - 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 8 - 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 8 - 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 8 - 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 2
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 6 - 2
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8 - 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 8 - 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 8 - 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8 - 2
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 8 - 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 8 - 2
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 6 - 2
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6 - 2
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 8 - 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 4 - 2
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001893
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001893: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 57%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001893: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [3,2,1] => [3,2,1] => 4
[3,1,2] => [3,2,1] => [3,2,1] => 4
[3,2,1] => [3,2,1] => [3,2,1] => 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 4
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 4
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 4
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 6
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 4
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 4
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 6
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => 6
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 6
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => 6
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 4
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 4
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 4
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 6
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 6
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 6
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 4
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 6
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 6
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 4
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 6
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 4
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 6
[3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 6
[3,4,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 6
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,2,1,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 6
[4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 6
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,3,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 6
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 6
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,5,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[4,5,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,2,3,1,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,3,1,2,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,3,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,3,2,1,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,3,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,3,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,4,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[5,4,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
Description
The flag descent of a signed permutation.
$$ fdes(\sigma) = 2 \lvert \{ i \in [n-1] \mid \sigma(i) > \sigma(i+1) \} \rvert + \chi( \sigma(1) < 0 ) $$
It has the same distribution as the flag excedance statistic.
Matching statistic: St001892
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001892: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 57%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001892: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [3,2,1] => [2,3,1] => [2,3,1] => 4
[3,1,2] => [3,2,1] => [2,3,1] => [2,3,1] => 4
[3,2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 4
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 4
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 4
[2,4,3,1] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 6
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 4
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 4
[3,4,1,2] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 6
[3,4,2,1] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 6
[4,2,1,3] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 6
[4,2,3,1] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 6
[4,3,1,2] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 6
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 4
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 4
[1,3,5,4,2] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 6
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 4
[1,4,5,2,3] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 6
[1,4,5,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 6
[1,5,3,2,4] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 6
[1,5,3,4,2] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 6
[1,5,4,2,3] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 6
[1,5,4,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 6
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 4
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 6
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 6
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 6
[2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 4
[2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 6
[2,4,3,1,5] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 6
[2,4,5,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[2,5,3,4,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[2,5,4,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[3,1,2,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 4
[3,1,2,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 6
[3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 4
[3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 6
[3,4,1,2,5] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 6
[3,4,2,1,5] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 6
[3,4,5,1,2] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[3,4,5,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[3,5,1,4,2] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[3,5,2,4,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[3,5,4,1,2] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[3,5,4,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,2,1,3,5] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 6
[4,2,3,1,5] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 6
[4,2,5,1,3] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,2,5,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,3,1,2,5] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 6
[4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 6
[4,3,5,1,2] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,3,5,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,5,1,2,3] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,5,1,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,5,2,1,3] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,5,2,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,5,3,1,2] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[4,5,3,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,2,3,1,4] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,2,3,4,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,2,4,1,3] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,2,4,3,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,3,1,2,4] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,3,1,4,2] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,3,2,1,4] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,3,2,4,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,3,4,1,2] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,3,4,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,4,1,2,3] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[5,4,1,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
Description
The flag excedance statistic of a signed permutation.
This is the number of negative entries plus twice the number of excedances of the signed permutation. That is,
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
where
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|$$
It has the same distribution as the flag descent statistic.
Matching statistic: St000656
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000656: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000656: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> ? = 0
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ? = 0
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 2
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ? = 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 2
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 2
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ? = 0
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 2
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 2
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 2
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 6
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 6
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ? = 0
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 2
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 2
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 2
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 6
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 6
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 2
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,17),(0,18),(1,11),(1,12),(2,53),(3,50),(4,33),(5,16),(5,65),(6,13),(6,61),(7,14),(7,62),(8,15),(8,51),(9,25),(9,63),(10,24),(10,52),(11,54),(12,22),(12,23),(12,54),(13,36),(14,35),(15,40),(16,59),(17,1),(17,48),(18,7),(18,48),(19,32),(19,45),(20,29),(20,38),(21,55),(21,58),(22,49),(22,60),(23,42),(23,49),(24,31),(24,46),(25,21),(25,60),(25,64),(27,70),(28,70),(29,67),(30,66),(31,68),(32,3),(32,69),(33,8),(34,39),(35,57),(36,39),(37,32),(37,66),(38,19),(38,37),(38,67),(39,26),(40,26),(41,30),(41,67),(42,52),(43,44),(43,68),(44,28),(44,69),(45,27),(45,69),(46,53),(46,68),(47,61),(48,9),(48,62),(49,5),(49,56),(50,34),(51,40),(52,2),(52,46),(53,6),(53,47),(54,10),(54,42),(55,31),(55,43),(56,43),(56,65),(57,29),(57,41),(58,30),(58,37),(59,27),(59,28),(60,55),(60,56),(61,34),(61,36),(62,35),(62,63),(63,20),(63,57),(63,64),(64,38),(64,41),(64,58),(65,44),(65,45),(65,59),(66,33),(67,4),(67,66),(68,47),(69,50),(69,70),(70,51)],71)
=> ? = 8
[2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,20),(0,21),(1,11),(2,10),(3,27),(3,36),(4,16),(4,34),(5,17),(5,35),(6,26),(6,52),(7,12),(7,53),(8,13),(8,55),(9,15),(9,25),(9,54),(10,51),(11,14),(11,63),(12,38),(13,39),(14,60),(15,19),(15,50),(16,18),(16,64),(17,56),(18,59),(19,22),(19,57),(20,9),(20,58),(21,8),(21,58),(22,47),(22,61),(23,31),(23,46),(24,45),(24,48),(25,37),(25,50),(26,24),(26,49),(26,62),(27,23),(27,61),(27,64),(29,68),(30,68),(31,67),(32,65),(33,69),(34,1),(35,2),(36,6),(37,36),(38,35),(39,34),(40,32),(40,67),(41,51),(41,69),(42,28),(43,28),(44,29),(45,41),(45,66),(46,63),(46,67),(47,52),(48,30),(48,66),(49,48),(49,65),(50,7),(50,57),(51,43),(52,62),(53,5),(53,38),(54,3),(54,37),(55,4),(55,39),(56,33),(56,41),(57,47),(57,53),(58,54),(58,55),(59,32),(59,49),(60,29),(60,30),(61,31),(61,40),(62,45),(62,56),(62,65),(63,44),(63,60),(64,40),(64,46),(64,59),(65,33),(65,66),(66,68),(66,69),(67,44),(68,42),(69,42),(69,43)],70)
=> ? = 8
[2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,21),(0,22),(0,44),(1,103),(2,102),(3,19),(3,105),(4,20),(4,104),(5,40),(5,60),(6,37),(6,101),(7,42),(7,129),(8,36),(8,132),(9,34),(9,126),(10,41),(10,127),(11,38),(11,108),(12,32),(12,107),(13,43),(13,59),(14,35),(14,100),(15,18),(15,33),(15,106),(16,39),(16,128),(17,119),(18,26),(18,27),(18,99),(19,17),(19,130),(20,113),(21,16),(21,97),(22,11),(22,116),(23,78),(23,120),(24,58),(24,74),(25,57),(25,70),(26,98),(26,124),(27,23),(27,98),(27,115),(28,63),(28,117),(29,64),(29,73),(30,93),(30,95),(31,111),(31,118),(32,50),(32,89),(33,88),(33,99),(34,85),(34,87),(35,30),(35,86),(35,122),(36,84),(36,125),(37,55),(37,96),(38,51),(38,90),(39,51),(39,91),(40,49),(40,92),(41,28),(41,121),(41,123),(42,31),(42,124),(42,131),(43,29),(43,120),(43,130),(44,15),(44,97),(44,116),(45,145),(46,135),(47,133),(48,146),(49,134),(50,141),(51,142),(52,139),(53,136),(54,136),(55,138),(56,147),(57,2),(57,143),(58,1),(58,140),(59,14),(60,6),(61,77),(62,49),(62,145),(63,75),(64,84),(64,144),(65,54),(66,57),(66,133),(67,65),(68,53),(69,53),(70,62),(70,143),(71,126),(72,127),(73,121),(73,144),(74,25),(74,66),(74,140),(75,94),(76,52),(76,146),(77,52),(77,134),(78,100),(79,85),(79,147),(80,81),(80,141),(81,45),(81,143),(82,46),(82,144),(83,47),(83,140),(84,61),(85,68),(86,95),(86,135),(87,65),(88,59),(89,104),(89,141),(90,105),(90,142),(91,110),(91,142),(92,94),(92,134),(93,79),(93,137),(94,55),(94,139),(95,48),(95,137),(96,54),(96,138),(97,7),(97,128),(98,8),(98,114),(99,12),(99,115),(100,122),(101,96),(102,67),(103,109),(104,9),(104,71),(105,10),(105,72),(106,13),(106,88),(107,4),(107,89),(108,3),(108,90),(109,75),(109,92),(110,58),(110,83),(111,50),(111,80),(112,76),(112,77),(113,56),(113,79),(114,80),(114,132),(115,78),(115,107),(116,106),(116,108),(117,48),(117,76),(118,47),(118,66),(119,46),(119,86),(120,64),(120,82),(121,112),(121,117),(122,93),(122,113),(122,135),(123,63),(123,109),(124,111),(124,114),(125,45),(125,62),(126,67),(126,87),(127,103),(127,123),(128,91),(128,129),(129,24),(129,110),(129,131),(130,73),(130,82),(130,119),(131,74),(131,83),(131,118),(132,70),(132,81),(132,125),(133,60),(134,139),(135,56),(135,137),(137,146),(137,147),(138,136),(139,138),(140,5),(140,133),(141,71),(142,72),(143,102),(143,145),(144,61),(144,112),(145,101),(146,69),(147,68),(147,69)],148)
=> ? = 8
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 6
[3,4,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,14),(1,22),(2,9),(3,1),(4,7),(4,11),(5,15),(5,91),(6,76),(7,27),(7,28),(7,96),(8,29),(8,75),(9,47),(10,25),(10,95),(11,24),(11,96),(12,26),(12,89),(13,16),(13,94),(14,17),(14,90),(15,51),(16,82),(17,81),(18,66),(18,84),(19,21),(19,67),(19,85),(20,45),(20,74),(21,48),(21,49),(22,60),(22,90),(23,41),(23,46),(24,60),(24,93),(25,56),(25,77),(26,69),(26,70),(27,42),(27,79),(28,42),(28,80),(29,34),(29,54),(30,39),(30,44),(30,65),(31,115),(32,115),(33,108),(34,97),(35,113),(35,114),(36,116),(37,109),(37,114),(38,117),(39,18),(39,109),(39,113),(40,100),(41,103),(42,13),(42,112),(43,8),(44,12),(44,113),(45,5),(45,101),(46,6),(46,103),(48,57),(48,99),(49,45),(49,99),(50,86),(50,98),(51,83),(52,63),(52,97),(53,75),(54,64),(54,97),(55,47),(56,59),(57,56),(57,111),(58,53),(58,111),(59,52),(60,78),(61,40),(61,116),(62,33),(62,107),(63,31),(63,110),(64,32),(64,110),(65,50),(65,109),(66,43),(67,20),(67,49),(67,117),(68,71),(68,105),(69,40),(69,106),(70,68),(70,106),(71,34),(71,102),(72,33),(72,102),(73,36),(73,104),(74,53),(74,101),(75,54),(76,55),(77,71),(77,72),(78,38),(78,85),(79,37),(79,65),(79,112),(80,35),(80,44),(80,112),(81,38),(81,67),(82,36),(82,61),(83,31),(83,32),(84,57),(84,58),(85,48),(85,84),(85,117),(86,61),(86,69),(86,104),(87,52),(87,92),(88,62),(88,72),(88,105),(89,43),(89,70),(90,19),(90,78),(90,81),(91,51),(91,92),(92,63),(92,64),(92,83),(93,35),(93,37),(93,39),(94,73),(94,82),(94,86),(95,68),(95,77),(95,88),(96,30),(96,79),(96,80),(96,93),(97,110),(98,95),(98,104),(99,101),(99,111),(100,107),(101,87),(101,91),(102,46),(102,108),(103,2),(104,88),(104,106),(104,116),(105,23),(105,102),(105,107),(106,100),(106,105),(107,41),(107,108),(108,103),(109,10),(109,98),(110,76),(110,115),(111,59),(111,87),(112,50),(112,94),(112,114),(113,66),(113,89),(114,73),(114,98),(115,55),(116,62),(116,100),(117,58),(117,74),(117,99)],118)
=> ? = 8
[3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,2),(1,45),(2,35),(3,37),(4,14),(4,49),(5,9),(5,48),(6,10),(6,12),(6,54),(7,31),(8,6),(8,13),(8,31),(9,27),(9,42),(10,28),(10,41),(11,26),(11,52),(12,17),(12,28),(12,51),(13,15),(13,50),(13,54),(14,16),(14,40),(14,53),(15,33),(15,44),(16,38),(16,47),(17,30),(17,40),(19,64),(19,66),(20,62),(21,65),(22,58),(23,60),(23,66),(24,60),(24,63),(25,67),(26,55),(27,59),(28,5),(28,56),(29,34),(30,22),(30,57),(31,4),(31,50),(32,22),(32,65),(33,43),(34,18),(35,18),(36,20),(37,1),(37,46),(38,27),(38,61),(39,19),(39,55),(39,58),(40,38),(40,57),(41,21),(41,56),(42,20),(42,59),(43,26),(43,39),(44,21),(44,32),(45,34),(45,35),(46,29),(46,45),(47,19),(47,23),(47,61),(48,36),(48,42),(49,11),(49,43),(49,53),(50,33),(50,49),(51,30),(51,32),(51,56),(52,23),(52,24),(52,55),(53,39),(53,47),(53,52),(53,57),(54,41),(54,44),(54,51),(55,3),(55,63),(55,66),(56,48),(56,65),(57,24),(57,58),(57,61),(58,63),(58,64),(59,25),(59,62),(60,25),(60,68),(61,59),(61,60),(61,64),(62,67),(63,68),(64,62),(64,68),(65,36),(66,37),(66,68),(67,29),(68,46),(68,67)],69)
=> ? = 8
[3,5,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,3,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,16),(0,17),(0,28),(1,9),(2,70),(3,13),(3,85),(4,27),(4,69),(5,83),(6,15),(6,43),(7,26),(7,84),(8,12),(8,29),(8,71),(9,68),(10,77),(11,76),(12,14),(12,78),(13,10),(13,86),(14,11),(14,87),(15,75),(16,8),(16,48),(17,7),(17,67),(18,42),(18,61),(19,25),(19,62),(19,63),(20,22),(20,59),(20,79),(21,24),(21,41),(21,60),(22,23),(22,80),(22,81),(23,64),(23,66),(24,37),(24,65),(25,36),(25,46),(26,50),(26,58),(27,31),(27,32),(28,48),(28,67),(29,57),(29,58),(29,78),(31,103),(32,103),(33,89),(34,89),(35,90),(36,88),(37,92),(38,102),(39,93),(40,101),(41,4),(41,100),(42,5),(42,97),(43,1),(44,53),(45,51),(45,95),(46,41),(46,88),(47,54),(48,71),(49,79),(50,73),(50,98),(51,68),(51,102),(52,31),(52,96),(53,30),(54,30),(55,33),(55,97),(56,35),(56,99),(57,70),(57,98),(58,74),(58,98),(59,81),(59,90),(60,37),(60,100),(61,34),(61,97),(62,36),(62,99),(63,21),(63,46),(63,99),(64,52),(64,91),(65,52),(65,92),(66,45),(66,91),(67,3),(67,84),(68,47),(69,32),(70,20),(70,49),(71,2),(71,57),(72,53),(72,54),(73,42),(73,55),(74,56),(74,62),(75,38),(75,51),(76,35),(76,59),(77,33),(77,34),(78,19),(78,74),(78,87),(79,80),(79,83),(79,90),(80,66),(80,82),(80,94),(81,64),(81,65),(81,94),(82,45),(82,75),(82,93),(83,39),(83,82),(84,50),(84,85),(85,18),(85,73),(85,86),(86,55),(86,61),(86,77),(87,56),(87,63),(87,76),(88,100),(89,43),(90,39),(90,94),(91,95),(91,96),(92,40),(92,96),(93,38),(93,40),(93,95),(94,91),(94,92),(94,93),(95,101),(95,102),(96,101),(96,103),(97,6),(97,89),(98,49),(99,60),(99,88),(100,69),(101,44),(101,72),(102,47),(102,72),(103,44)],104)
=> ? = 8
[3,5,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,12),(0,13),(1,6),(2,61),(3,5),(3,94),(4,81),(5,80),(6,32),(7,4),(7,95),(8,22),(8,87),(9,20),(9,85),(10,19),(10,92),(11,17),(11,18),(11,98),(12,51),(13,11),(13,21),(13,51),(14,46),(14,47),(15,42),(15,89),(16,73),(16,75),(17,27),(17,48),(17,97),(18,48),(18,90),(19,43),(19,91),(20,44),(20,76),(21,25),(21,88),(21,98),(22,26),(22,74),(22,96),(23,24),(23,45),(23,77),(24,66),(24,67),(25,53),(25,93),(26,68),(26,84),(27,74),(27,78),(28,37),(28,60),(28,72),(29,124),(30,124),(31,101),(31,123),(33,117),(33,125),(34,118),(35,99),(36,116),(36,128),(37,101),(37,121),(38,109),(38,125),(39,100),(39,103),(40,109),(40,129),(41,126),(42,119),(43,107),(44,113),(45,9),(45,106),(46,108),(47,2),(47,108),(48,10),(48,122),(49,65),(49,128),(50,64),(51,8),(51,88),(52,39),(52,121),(52,123),(53,79),(54,45),(54,100),(55,32),(56,29),(56,127),(57,30),(57,127),(58,35),(58,120),(59,34),(59,99),(60,63),(60,101),(61,55),(62,56),(62,104),(63,83),(63,111),(64,57),(64,104),(65,44),(65,114),(66,58),(66,110),(67,65),(67,110),(68,43),(68,115),(69,34),(69,105),(70,36),(70,112),(71,33),(71,103),(71,119),(72,23),(72,54),(72,121),(73,50),(74,68),(74,102),(75,3),(75,82),(76,64),(76,113),(77,66),(77,106),(78,39),(78,54),(78,102),(79,42),(79,71),(80,29),(80,30),(81,35),(81,59),(82,62),(82,94),(83,49),(83,67),(83,112),(84,33),(84,38),(84,115),(85,50),(85,76),(86,59),(86,69),(86,120),(87,15),(87,79),(87,96),(88,53),(88,87),(89,38),(89,40),(89,119),(90,31),(90,60),(90,122),(91,36),(91,49),(91,107),(92,70),(92,83),(92,91),(93,31),(93,37),(93,52),(94,56),(94,57),(94,80),(95,58),(95,81),(95,86),(96,71),(96,84),(96,89),(96,102),(97,52),(97,72),(97,78),(97,122),(98,28),(98,90),(98,93),(98,97),(99,46),(99,118),(100,106),(101,7),(101,111),(102,40),(102,103),(102,115),(103,117),(103,129),(104,127),(105,47),(105,118),(106,85),(107,41),(107,116),(108,1),(109,41),(109,130),(110,114),(110,120),(111,95),(111,112),(112,86),(112,110),(112,128),(113,104),(114,105),(114,113),(115,107),(115,109),(115,117),(116,126),(117,116),(117,130),(118,108),(119,16),(119,125),(119,129),(120,14),(120,99),(120,105),(121,77),(121,100),(122,63),(122,92),(122,123),(123,70),(123,111),(124,55),(125,75),(125,130),(126,62),(127,61),(127,124),(128,69),(128,114),(129,73),(129,130),(130,82),(130,126)],131)
=> ? = 8
[3,5,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,19),(0,20),(0,50),(1,40),(2,39),(3,169),(4,146),(5,124),(6,123),(7,8),(7,52),(7,125),(8,10),(8,180),(9,167),(10,9),(10,197),(11,22),(11,165),(12,35),(12,179),(13,31),(13,178),(14,26),(14,176),(15,25),(15,195),(16,30),(16,194),(17,27),(17,199),(18,28),(18,29),(18,202),(19,7),(19,102),(20,18),(20,32),(20,101),(21,82),(21,83),(22,74),(22,155),(23,75),(23,185),(24,181),(24,191),(25,148),(25,188),(26,73),(26,154),(27,147),(27,183),(28,49),(28,84),(28,201),(29,84),(29,187),(30,77),(30,190),(31,76),(31,156),(32,53),(32,184),(32,202),(33,45),(33,157),(33,186),(34,43),(34,151),(34,192),(35,46),(35,152),(35,200),(36,47),(36,153),(36,182),(37,41),(37,78),(37,189),(38,44),(38,79),(38,193),(39,80),(39,81),(40,42),(40,149),(40,150),(41,98),(41,99),(42,134),(42,135),(43,97),(43,166),(44,132),(44,133),(45,96),(45,100),(46,136),(46,174),(47,48),(47,171),(47,173),(48,126),(48,129),(49,152),(49,161),(50,101),(50,102),(51,72),(51,122),(51,145),(52,103),(52,109),(52,180),(53,109),(53,110),(53,198),(54,249),(55,248),(56,209),(57,208),(58,239),(59,237),(59,250),(60,238),(61,236),(61,255),(62,207),(62,247),(63,212),(64,211),(65,11),(65,220),(66,222),(66,250),(67,225),(67,226),(68,258),(69,220),(70,222),(70,223),(71,251),(72,24),(72,205),(72,207),(73,253),(74,204),(75,3),(75,240),(76,230),(77,217),(78,14),(78,216),(79,13),(79,229),(80,218),(81,218),(82,6),(82,221),(83,5),(83,221),(84,16),(84,246),(85,119),(85,204),(86,131),(86,255),(87,67),(87,205),(87,247),(88,64),(88,249),(89,63),(89,248),(90,142),(91,92),(91,241),(92,74),(92,203),(93,143),(94,81),(95,80),(96,104),(96,206),(97,113),(98,112),(98,256),(99,130),(99,256),(100,78),(100,206),(101,12),(101,184),(102,125),(103,146),(103,244),(104,147),(104,243),(105,182),(106,85),(107,89),(107,252),(108,88),(108,242),(109,158),(109,244),(110,163),(110,244),(111,160),(111,243),(112,115),(113,149),(114,148),(114,254),(115,150),(116,58),(116,203),(117,56),(117,257),(118,55),(118,252),(119,54),(119,242),(120,56),(120,254),(121,57),(121,245),(122,127),(122,207),(123,95),(124,94),(125,4),(125,103),(126,138),(126,235),(127,172),(127,228),(128,118),(128,214),(129,130),(129,235),(130,73),(130,213),(131,76),(131,233),(132,131),(132,227),(133,91),(133,227),(134,63),(134,232),(135,64),(135,232),(136,77),(136,234),(137,107),(137,214),(138,114),(138,219),(139,79),(139,226),(140,58),(140,215),(141,61),(141,224),(142,60),(142,211),(143,60),(143,212),(144,59),(144,225),(144,240),(145,38),(145,139),(145,205),(146,36),(146,105),(147,106),(148,93),(149,134),(149,209),(150,135),(150,209),(151,15),(151,170),(152,136),(152,210),(153,171),(153,208),(154,168),(154,253),(155,108),(155,204),(156,137),(156,230),(157,37),(157,100),(157,245),(158,121),(158,186),(159,113),(159,137),(160,112),(160,165),(161,67),(161,139),(161,210),(162,142),(162,143),(163,75),(163,144),(164,138),(164,166),(164,258),(165,115),(165,155),(166,114),(166,120),(167,57),(167,153),(168,54),(168,88),(169,68),(169,164),(170,128),(170,195),(171,99),(171,129),(171,231),(172,86),(172,132),(172,224),(173,126),(173,164),(173,231),(174,59),(174,66),(174,234),(175,116),(175,140),(175,241),(176,154),(176,196),(177,85),(177,196),(178,156),(178,159),(179,23),(179,163),(179,200),(180,33),(180,158),(180,197),(181,104),(181,111),(182,169),(182,173),(182,208),(183,92),(183,116),(184,110),(184,179),(185,66),(185,70),(185,240),(186,96),(186,181),(186,245),(187,62),(187,122),(187,246),(188,55),(188,89),(189,98),(189,160),(189,216),(190,61),(190,86),(190,217),(191,65),(191,69),(192,97),(192,159),(193,65),(193,133),(193,229),(194,141),(194,172),(194,190),(195,107),(195,118),(195,188),(196,108),(196,119),(196,168),(197,121),(197,157),(197,167),(198,62),(198,72),(198,87),(199,91),(199,175),(199,183),(200,144),(200,174),(200,185),(200,210),(201,87),(201,145),(201,161),(201,246),(202,51),(202,187),(202,198),(202,201),(203,83),(203,239),(204,242),(205,191),(205,193),(205,226),(206,216),(206,243),(207,17),(207,228),(208,68),(208,231),(209,232),(210,70),(210,225),(210,234),(211,238),(212,238),(213,253),(213,257),(214,252),(215,82),(215,239),(216,176),(216,177),(217,71),(217,236),(219,254),(219,257),(220,1),(221,2),(222,71),(222,259),(223,192),(223,259),(224,175),(224,227),(224,255),(225,223),(225,237),(226,69),(226,229),(227,233),(227,241),(228,199),(228,224),(229,178),(229,220),(230,214),(231,235),(231,256),(231,258),(232,211),(232,212),(233,215),(233,230),(234,217),(234,222),(234,237),(235,213),(235,219),(236,251),(237,236),(237,259),(238,218),(239,221),(240,34),(240,223),(240,250),(241,21),(241,203),(241,215),(242,124),(242,249),(243,106),(243,177),(244,105),(245,111),(245,189),(245,206),(246,127),(246,194),(246,247),(247,141),(247,228),(248,95),(249,94),(250,151),(250,259),(251,128),(252,123),(252,248),(253,90),(254,93),(254,162),(255,140),(255,233),(256,117),(256,213),(257,90),(257,162),(258,117),(258,120),(258,219),(259,170),(259,251)],260)
=> ? = 8
[4,2,1,3,5] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 6
[4,2,5,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,2),(1,45),(2,35),(3,37),(4,14),(4,49),(5,9),(5,48),(6,10),(6,12),(6,54),(7,31),(8,6),(8,13),(8,31),(9,27),(9,42),(10,28),(10,41),(11,26),(11,52),(12,17),(12,28),(12,51),(13,15),(13,50),(13,54),(14,16),(14,40),(14,53),(15,33),(15,44),(16,38),(16,47),(17,30),(17,40),(19,64),(19,66),(20,62),(21,65),(22,58),(23,60),(23,66),(24,60),(24,63),(25,67),(26,55),(27,59),(28,5),(28,56),(29,34),(30,22),(30,57),(31,4),(31,50),(32,22),(32,65),(33,43),(34,18),(35,18),(36,20),(37,1),(37,46),(38,27),(38,61),(39,19),(39,55),(39,58),(40,38),(40,57),(41,21),(41,56),(42,20),(42,59),(43,26),(43,39),(44,21),(44,32),(45,34),(45,35),(46,29),(46,45),(47,19),(47,23),(47,61),(48,36),(48,42),(49,11),(49,43),(49,53),(50,33),(50,49),(51,30),(51,32),(51,56),(52,23),(52,24),(52,55),(53,39),(53,47),(53,52),(53,57),(54,41),(54,44),(54,51),(55,3),(55,63),(55,66),(56,48),(56,65),(57,24),(57,58),(57,61),(58,63),(58,64),(59,25),(59,62),(60,25),(60,68),(61,59),(61,60),(61,64),(62,67),(63,68),(64,62),(64,68),(65,36),(66,37),(66,68),(67,29),(68,46),(68,67)],69)
=> ? = 8
[4,2,5,3,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,10),(1,11),(2,87),(3,13),(3,113),(4,89),(5,31),(5,107),(6,25),(6,108),(7,27),(7,29),(7,114),(8,24),(8,88),(9,12),(9,115),(10,90),(11,26),(11,28),(11,90),(12,112),(13,93),(14,1),(14,86),(15,7),(15,20),(15,86),(16,70),(16,71),(17,83),(17,84),(18,69),(18,99),(19,22),(19,82),(19,104),(20,23),(20,98),(20,114),(21,42),(21,100),(22,74),(22,75),(23,62),(23,95),(24,55),(24,58),(25,54),(25,56),(26,57),(26,85),(27,47),(27,66),(28,85),(28,109),(29,18),(29,47),(29,92),(30,81),(30,97),(30,103),(31,30),(31,99),(31,109),(31,110),(32,138),(33,147),(34,146),(35,117),(36,117),(36,147),(37,140),(37,148),(38,139),(39,137),(40,136),(41,150),(42,142),(43,125),(44,129),(45,135),(46,127),(46,151),(47,6),(47,116),(48,71),(48,148),(48,151),(49,64),(50,77),(50,152),(51,73),(51,150),(52,43),(53,65),(54,96),(54,153),(55,76),(55,118),(56,43),(56,153),(57,88),(58,89),(58,118),(59,67),(59,149),(60,68),(60,118),(60,149),(61,59),(61,144),(62,94),(63,44),(63,146),(64,32),(64,133),(65,38),(65,132),(66,34),(66,116),(67,35),(67,145),(68,36),(68,143),(68,145),(69,44),(69,119),(70,33),(70,143),(71,17),(71,91),(71,143),(72,78),(72,130),(73,45),(73,123),(74,32),(74,121),(75,40),(75,121),(76,41),(76,126),(77,39),(77,123),(78,40),(78,124),(79,38),(79,124),(80,37),(80,129),(80,142),(81,54),(81,122),(81,144),(82,75),(82,131),(83,72),(83,128),(84,39),(84,128),(85,9),(85,111),(86,5),(86,98),(87,19),(87,101),(88,4),(88,58),(89,3),(89,106),(90,8),(90,57),(91,77),(91,84),(91,120),(92,63),(92,69),(92,116),(93,78),(93,79),(94,42),(94,80),(95,34),(95,63),(96,41),(96,51),(97,55),(97,60),(97,144),(98,62),(98,107),(99,61),(99,81),(99,119),(100,16),(100,46),(100,48),(100,142),(101,49),(101,104),(102,65),(102,79),(102,130),(103,37),(103,48),(103,122),(104,64),(104,74),(104,131),(105,53),(105,102),(106,105),(106,113),(107,21),(107,94),(107,110),(108,52),(108,56),(109,61),(109,97),(109,111),(110,80),(110,100),(110,103),(110,119),(111,59),(111,60),(111,115),(112,33),(112,35),(112,36),(113,72),(113,93),(113,102),(114,66),(114,92),(114,95),(115,67),(115,68),(115,70),(115,112),(116,108),(116,146),(117,45),(117,134),(118,106),(118,126),(119,46),(119,122),(119,129),(120,123),(120,128),(120,134),(121,136),(121,138),(122,140),(122,151),(122,153),(123,135),(123,137),(124,136),(124,139),(125,152),(126,105),(126,150),(127,154),(128,130),(128,137),(129,127),(129,140),(130,124),(130,132),(131,121),(131,133),(132,139),(133,138),(134,131),(134,135),(135,133),(136,141),(137,132),(138,141),(139,141),(140,125),(140,154),(142,2),(142,127),(142,148),(143,83),(143,120),(143,147),(144,76),(144,96),(144,149),(145,73),(145,117),(145,120),(146,52),(147,82),(147,134),(148,87),(148,154),(149,51),(149,126),(149,145),(150,53),(151,50),(151,91),(151,154),(152,49),(153,50),(153,125),(154,101),(154,152)],155)
=> ? = 8
[4,3,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6
[4,3,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,2),(0,3),(1,8),(1,16),(2,1),(3,7),(3,10),(4,13),(4,59),(5,12),(5,81),(6,20),(6,82),(7,11),(7,83),(8,60),(9,23),(9,58),(10,19),(10,22),(10,83),(11,78),(12,64),(13,24),(14,52),(14,72),(15,17),(15,73),(15,80),(16,60),(16,74),(17,39),(17,70),(18,36),(18,44),(19,51),(19,69),(20,33),(20,75),(21,28),(21,38),(21,50),(22,51),(22,71),(22,74),(23,49),(23,56),(23,57),(25,101),(26,113),(27,107),(27,110),(28,107),(28,111),(29,110),(29,111),(30,112),(31,90),(31,98),(32,94),(32,114),(33,96),(34,91),(34,92),(35,89),(35,92),(36,108),(37,94),(37,103),(38,9),(38,111),(39,45),(40,31),(40,102),(40,114),(41,31),(41,104),(42,37),(42,99),(43,34),(43,85),(43,87),(44,30),(44,108),(45,55),(46,33),(46,86),(47,48),(47,86),(48,25),(48,100),(49,42),(49,85),(50,14),(50,76),(50,107),(51,15),(51,61),(51,84),(52,46),(52,109),(53,32),(53,93),(54,26),(54,95),(55,25),(55,91),(56,35),(56,85),(56,97),(57,53),(57,97),(58,57),(59,24),(60,5),(60,63),(61,67),(61,73),(62,34),(62,35),(62,113),(63,67),(63,81),(64,46),(64,47),(65,45),(65,68),(66,47),(66,68),(66,109),(67,65),(67,66),(68,48),(68,55),(68,87),(69,29),(69,38),(69,84),(70,26),(70,62),(71,27),(71,50),(71,84),(72,43),(72,49),(72,109),(73,39),(73,65),(74,61),(74,63),(75,32),(75,40),(75,96),(76,72),(76,79),(76,88),(77,40),(77,41),(77,93),(78,27),(78,28),(78,29),(79,43),(79,56),(79,62),(79,95),(80,54),(80,70),(80,79),(81,52),(81,64),(81,66),(82,53),(82,75),(82,77),(83,21),(83,69),(83,71),(83,78),(84,76),(84,80),(84,110),(85,92),(85,99),(86,37),(86,96),(86,100),(87,91),(87,99),(87,100),(88,82),(88,95),(89,104),(90,108),(90,116),(91,101),(91,106),(92,106),(93,18),(93,104),(93,114),(94,115),(95,77),(95,97),(95,113),(96,94),(96,102),(97,89),(97,93),(98,116),(99,103),(99,106),(100,101),(100,102),(100,103),(101,98),(101,105),(102,98),(102,115),(103,105),(103,115),(104,36),(104,90),(105,116),(106,105),(107,6),(107,88),(108,4),(108,112),(109,42),(109,86),(109,87),(110,54),(110,88),(111,58),(112,59),(113,41),(113,89),(114,44),(114,90),(114,115),(115,30),(115,116),(116,112)],117)
=> ? = 8
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 2
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 2
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 2
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 2
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 2
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
Description
The number of cuts of a poset.
A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.
Matching statistic: St001880
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001880: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001880: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> ? = 0
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ? = 0
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ? = 2
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ? = 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> ? = 2
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> ? = 2
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ? = 0
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> ? = 2
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> ? = 2
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> ? = 2
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 6
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 6
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ? = 0
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> ? = 2
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> ? = 2
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ? = 2
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 6
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 6
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ? = 2
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? = 6
[2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,17),(0,18),(1,11),(1,12),(2,53),(3,50),(4,33),(5,16),(5,65),(6,13),(6,61),(7,14),(7,62),(8,15),(8,51),(9,25),(9,63),(10,24),(10,52),(11,54),(12,22),(12,23),(12,54),(13,36),(14,35),(15,40),(16,59),(17,1),(17,48),(18,7),(18,48),(19,32),(19,45),(20,29),(20,38),(21,55),(21,58),(22,49),(22,60),(23,42),(23,49),(24,31),(24,46),(25,21),(25,60),(25,64),(27,70),(28,70),(29,67),(30,66),(31,68),(32,3),(32,69),(33,8),(34,39),(35,57),(36,39),(37,32),(37,66),(38,19),(38,37),(38,67),(39,26),(40,26),(41,30),(41,67),(42,52),(43,44),(43,68),(44,28),(44,69),(45,27),(45,69),(46,53),(46,68),(47,61),(48,9),(48,62),(49,5),(49,56),(50,34),(51,40),(52,2),(52,46),(53,6),(53,47),(54,10),(54,42),(55,31),(55,43),(56,43),(56,65),(57,29),(57,41),(58,30),(58,37),(59,27),(59,28),(60,55),(60,56),(61,34),(61,36),(62,35),(62,63),(63,20),(63,57),(63,64),(64,38),(64,41),(64,58),(65,44),(65,45),(65,59),(66,33),(67,4),(67,66),(68,47),(69,50),(69,70),(70,51)],71)
=> ? = 8
[2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,20),(0,21),(1,11),(2,10),(3,27),(3,36),(4,16),(4,34),(5,17),(5,35),(6,26),(6,52),(7,12),(7,53),(8,13),(8,55),(9,15),(9,25),(9,54),(10,51),(11,14),(11,63),(12,38),(13,39),(14,60),(15,19),(15,50),(16,18),(16,64),(17,56),(18,59),(19,22),(19,57),(20,9),(20,58),(21,8),(21,58),(22,47),(22,61),(23,31),(23,46),(24,45),(24,48),(25,37),(25,50),(26,24),(26,49),(26,62),(27,23),(27,61),(27,64),(29,68),(30,68),(31,67),(32,65),(33,69),(34,1),(35,2),(36,6),(37,36),(38,35),(39,34),(40,32),(40,67),(41,51),(41,69),(42,28),(43,28),(44,29),(45,41),(45,66),(46,63),(46,67),(47,52),(48,30),(48,66),(49,48),(49,65),(50,7),(50,57),(51,43),(52,62),(53,5),(53,38),(54,3),(54,37),(55,4),(55,39),(56,33),(56,41),(57,47),(57,53),(58,54),(58,55),(59,32),(59,49),(60,29),(60,30),(61,31),(61,40),(62,45),(62,56),(62,65),(63,44),(63,60),(64,40),(64,46),(64,59),(65,33),(65,66),(66,68),(66,69),(67,44),(68,42),(69,42),(69,43)],70)
=> ? = 8
[2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,21),(0,22),(0,44),(1,103),(2,102),(3,19),(3,105),(4,20),(4,104),(5,40),(5,60),(6,37),(6,101),(7,42),(7,129),(8,36),(8,132),(9,34),(9,126),(10,41),(10,127),(11,38),(11,108),(12,32),(12,107),(13,43),(13,59),(14,35),(14,100),(15,18),(15,33),(15,106),(16,39),(16,128),(17,119),(18,26),(18,27),(18,99),(19,17),(19,130),(20,113),(21,16),(21,97),(22,11),(22,116),(23,78),(23,120),(24,58),(24,74),(25,57),(25,70),(26,98),(26,124),(27,23),(27,98),(27,115),(28,63),(28,117),(29,64),(29,73),(30,93),(30,95),(31,111),(31,118),(32,50),(32,89),(33,88),(33,99),(34,85),(34,87),(35,30),(35,86),(35,122),(36,84),(36,125),(37,55),(37,96),(38,51),(38,90),(39,51),(39,91),(40,49),(40,92),(41,28),(41,121),(41,123),(42,31),(42,124),(42,131),(43,29),(43,120),(43,130),(44,15),(44,97),(44,116),(45,145),(46,135),(47,133),(48,146),(49,134),(50,141),(51,142),(52,139),(53,136),(54,136),(55,138),(56,147),(57,2),(57,143),(58,1),(58,140),(59,14),(60,6),(61,77),(62,49),(62,145),(63,75),(64,84),(64,144),(65,54),(66,57),(66,133),(67,65),(68,53),(69,53),(70,62),(70,143),(71,126),(72,127),(73,121),(73,144),(74,25),(74,66),(74,140),(75,94),(76,52),(76,146),(77,52),(77,134),(78,100),(79,85),(79,147),(80,81),(80,141),(81,45),(81,143),(82,46),(82,144),(83,47),(83,140),(84,61),(85,68),(86,95),(86,135),(87,65),(88,59),(89,104),(89,141),(90,105),(90,142),(91,110),(91,142),(92,94),(92,134),(93,79),(93,137),(94,55),(94,139),(95,48),(95,137),(96,54),(96,138),(97,7),(97,128),(98,8),(98,114),(99,12),(99,115),(100,122),(101,96),(102,67),(103,109),(104,9),(104,71),(105,10),(105,72),(106,13),(106,88),(107,4),(107,89),(108,3),(108,90),(109,75),(109,92),(110,58),(110,83),(111,50),(111,80),(112,76),(112,77),(113,56),(113,79),(114,80),(114,132),(115,78),(115,107),(116,106),(116,108),(117,48),(117,76),(118,47),(118,66),(119,46),(119,86),(120,64),(120,82),(121,112),(121,117),(122,93),(122,113),(122,135),(123,63),(123,109),(124,111),(124,114),(125,45),(125,62),(126,67),(126,87),(127,103),(127,123),(128,91),(128,129),(129,24),(129,110),(129,131),(130,73),(130,82),(130,119),(131,74),(131,83),(131,118),(132,70),(132,81),(132,125),(133,60),(134,139),(135,56),(135,137),(137,146),(137,147),(138,136),(139,138),(140,5),(140,133),(141,71),(142,72),(143,102),(143,145),(144,61),(144,112),(145,101),(146,69),(147,68),(147,69)],148)
=> ? = 8
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4
[3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 6
[3,4,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 6
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,14),(1,22),(2,9),(3,1),(4,7),(4,11),(5,15),(5,91),(6,76),(7,27),(7,28),(7,96),(8,29),(8,75),(9,47),(10,25),(10,95),(11,24),(11,96),(12,26),(12,89),(13,16),(13,94),(14,17),(14,90),(15,51),(16,82),(17,81),(18,66),(18,84),(19,21),(19,67),(19,85),(20,45),(20,74),(21,48),(21,49),(22,60),(22,90),(23,41),(23,46),(24,60),(24,93),(25,56),(25,77),(26,69),(26,70),(27,42),(27,79),(28,42),(28,80),(29,34),(29,54),(30,39),(30,44),(30,65),(31,115),(32,115),(33,108),(34,97),(35,113),(35,114),(36,116),(37,109),(37,114),(38,117),(39,18),(39,109),(39,113),(40,100),(41,103),(42,13),(42,112),(43,8),(44,12),(44,113),(45,5),(45,101),(46,6),(46,103),(48,57),(48,99),(49,45),(49,99),(50,86),(50,98),(51,83),(52,63),(52,97),(53,75),(54,64),(54,97),(55,47),(56,59),(57,56),(57,111),(58,53),(58,111),(59,52),(60,78),(61,40),(61,116),(62,33),(62,107),(63,31),(63,110),(64,32),(64,110),(65,50),(65,109),(66,43),(67,20),(67,49),(67,117),(68,71),(68,105),(69,40),(69,106),(70,68),(70,106),(71,34),(71,102),(72,33),(72,102),(73,36),(73,104),(74,53),(74,101),(75,54),(76,55),(77,71),(77,72),(78,38),(78,85),(79,37),(79,65),(79,112),(80,35),(80,44),(80,112),(81,38),(81,67),(82,36),(82,61),(83,31),(83,32),(84,57),(84,58),(85,48),(85,84),(85,117),(86,61),(86,69),(86,104),(87,52),(87,92),(88,62),(88,72),(88,105),(89,43),(89,70),(90,19),(90,78),(90,81),(91,51),(91,92),(92,63),(92,64),(92,83),(93,35),(93,37),(93,39),(94,73),(94,82),(94,86),(95,68),(95,77),(95,88),(96,30),(96,79),(96,80),(96,93),(97,110),(98,95),(98,104),(99,101),(99,111),(100,107),(101,87),(101,91),(102,46),(102,108),(103,2),(104,88),(104,106),(104,116),(105,23),(105,102),(105,107),(106,100),(106,105),(107,41),(107,108),(108,103),(109,10),(109,98),(110,76),(110,115),(111,59),(111,87),(112,50),(112,94),(112,114),(113,66),(113,89),(114,73),(114,98),(115,55),(116,62),(116,100),(117,58),(117,74),(117,99)],118)
=> ? = 8
[3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,2),(1,45),(2,35),(3,37),(4,14),(4,49),(5,9),(5,48),(6,10),(6,12),(6,54),(7,31),(8,6),(8,13),(8,31),(9,27),(9,42),(10,28),(10,41),(11,26),(11,52),(12,17),(12,28),(12,51),(13,15),(13,50),(13,54),(14,16),(14,40),(14,53),(15,33),(15,44),(16,38),(16,47),(17,30),(17,40),(19,64),(19,66),(20,62),(21,65),(22,58),(23,60),(23,66),(24,60),(24,63),(25,67),(26,55),(27,59),(28,5),(28,56),(29,34),(30,22),(30,57),(31,4),(31,50),(32,22),(32,65),(33,43),(34,18),(35,18),(36,20),(37,1),(37,46),(38,27),(38,61),(39,19),(39,55),(39,58),(40,38),(40,57),(41,21),(41,56),(42,20),(42,59),(43,26),(43,39),(44,21),(44,32),(45,34),(45,35),(46,29),(46,45),(47,19),(47,23),(47,61),(48,36),(48,42),(49,11),(49,43),(49,53),(50,33),(50,49),(51,30),(51,32),(51,56),(52,23),(52,24),(52,55),(53,39),(53,47),(53,52),(53,57),(54,41),(54,44),(54,51),(55,3),(55,63),(55,66),(56,48),(56,65),(57,24),(57,58),(57,61),(58,63),(58,64),(59,25),(59,62),(60,25),(60,68),(61,59),(61,60),(61,64),(62,67),(63,68),(64,62),(64,68),(65,36),(66,37),(66,68),(67,29),(68,46),(68,67)],69)
=> ? = 8
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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