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Your data matches 31 different statistics following compositions of up to 3 maps.
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Matching statistic: St001060
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(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,4,5,3,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,4,2,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[2,1,4,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[2,1,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[2,3,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,3,4,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[2,4,1,5,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,1,4,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[2,5,3,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[3,1,4,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[3,1,4,5,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[3,1,5,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[3,2,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,5,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[3,4,1,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,4,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[4,1,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[4,1,5,2,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[4,1,5,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[4,2,1,5,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[4,2,3,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000264
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 52%●distinct values known / distinct values provided: 33%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 52%●distinct values known / distinct values provided: 33%
Values
[1,3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,3,4] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[2,3,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,4,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,4,2,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,4,3,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[2,1,3,4,5] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,1,3,5,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,1,5,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[2,1,5,4,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,3,1,5,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[2,3,4,1,5] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[2,4,1,3,5] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,4,1,5,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,3,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,4,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,3,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,4,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,4,5,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 1
[3,1,5,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,5,4,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[3,2,1,4,5] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 1
[3,2,1,5,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[3,2,4,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,5,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,4,1,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[3,4,2,1,5] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[4,1,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,5,2,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[4,1,5,3,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[4,2,1,3,5] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,1,5,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[4,2,3,1,5] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[4,3,1,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[4,3,2,1,5] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 + 1
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,6,4,2] => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,2,3,6] => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,3,6,2] => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001570
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%
Values
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[1,4,5,3,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[1,5,2,4,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,5,3,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,5,3,4,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[1,5,4,2,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,4,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[2,1,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[2,3,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[2,3,4,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[2,4,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,4,1,5,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,4,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,5,3,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,1,4,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,1,4,5,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[3,1,5,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,5,4,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[3,2,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,2,5,1,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,4,1,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[3,4,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[4,1,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[4,1,5,2,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[4,1,5,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[4,2,1,5,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[4,2,3,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[4,3,1,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[4,3,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 3 - 2
[1,3,4,5,6,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 4 - 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,6,2,5,3] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,6,3,2,5] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,5,2,4,6,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,5,3,2,4,6] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,5,6,3,2,4] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,2,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,2,4,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,2,4,5,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,2,5,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,2,5,4,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,3,2,4,5] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,3,2,5,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,3,4,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,3,5,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,3,5,4,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,4,2,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,4,2,5,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,4,3,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,4,3,5,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,5,2,4,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,6,5,3,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,1,6,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,1,6,4,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,5,1,6,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,5,4,1,6] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,1,5,6,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,1,6,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,1,6,5,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,3,1,5,6] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,3,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,3,5,1,6] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,5,3,1,6] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,6,1,5,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,6,3,1,5] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,3,6,4] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,4,6,3] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,6,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,6,4,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,3,1,4,6] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,3,1,6,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,5,2,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,5,2,4,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,3,2,4] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,3,4,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,5,4,2,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,5,4,3,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[2,1,3,4,5] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[2,1,3,5,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[2,1,5,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[2,1,5,4,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[2,3,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[2,3,1,5,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[2,4,1,3,5] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,4,3,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,5,1,4,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,5,3,1,4] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,1,4,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,1,4,5,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[3,1,5,2,4] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,1,5,4,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[3,2,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[3,2,1,5,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[3,2,4,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,2,5,1,4] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,4,1,5,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[3,4,2,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[4,1,3,5,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[4,1,5,2,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[4,1,5,3,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[4,2,1,3,5] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[4,2,1,5,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[4,2,3,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,3,5,1,4] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,4,1,5,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,4,3,1,5] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,5,1,4,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,5,3,1,4] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,4,2,6,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,5,2,6,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,5,4,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,6,2,5,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,6,4,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,6,5,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,2,4,1,6,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,2,5,1,6,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,2,5,4,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,2,6,1,5,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000422
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 4
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[1,5,2,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[1,5,2,4,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,5,3,2,4] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,5,3,4,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[1,5,4,2,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[1,5,4,3,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[2,1,3,4,5] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 4
[2,1,3,5,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[2,1,4,5,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[2,1,5,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[2,1,5,4,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[2,3,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[2,3,1,5,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[2,4,1,3,5] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[2,4,1,5,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[2,4,3,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[2,5,1,4,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[2,5,3,1,4] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[3,1,4,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[3,1,4,5,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[3,1,5,2,4] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[3,1,5,4,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[3,2,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[3,2,1,5,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[3,2,4,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[3,2,5,1,4] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[3,4,1,5,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[3,4,2,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[4,1,3,5,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[4,1,5,2,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[4,1,5,3,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[4,2,1,3,5] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[4,2,1,5,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[4,2,3,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,3,5,1,4] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,4,1,5,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,4,3,1,5] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,5,1,4,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,6,5,3,1,4] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,4,2,6,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,5,2,6,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,5,4,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,6,2,5,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,6,4,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,6,5,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,2,4,1,6,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,2,5,1,6,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,2,5,4,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,2,6,1,5,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
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: St001616
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2
[2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2
[3,2,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[2,4,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2
[3,1,4,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[3,2,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[3,4,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2
[4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[4,2,1,3,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[4,2,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[4,3,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,3,4,5,6,2] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 4
[1,3,4,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[1,3,4,6,5,2] => [1,6,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 2
[1,3,5,2,6,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[1,3,5,4,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 2
[1,3,5,6,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2
[1,3,5,6,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2
[1,3,6,2,4,5] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[1,3,6,4,2,5] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2
[1,3,6,4,5,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2
[1,3,6,5,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2
[1,3,6,5,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2
[1,4,2,5,6,3] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 2
[1,4,2,6,3,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[2,3,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[2,3,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,3,6,1,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,3,6,4,1,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,4,1,6,3,5] => [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,4,1,6,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,5,1,3,4,6] => [3,5,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,5,1,4,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,5,3,1,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,5,4,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,6,1,3,5,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,1,4,3,5] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,1,4,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,1,5,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,1,5,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,3,1,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,6,3,1,5,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[3,1,5,2,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[3,1,5,2,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,6,2,4,5] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,6,2,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,6,4,2,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,6,5,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,2,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[3,2,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[3,2,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
Description
The number of neutral elements in a lattice.
An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St001720
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2
[2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2
[3,2,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[2,4,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2
[3,1,4,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3
[3,2,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[3,4,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2
[4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[4,2,1,3,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[4,2,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
[4,3,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3
[1,3,4,5,6,2] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 4
[1,3,4,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[1,3,4,6,5,2] => [1,6,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 2
[1,3,5,2,6,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[1,3,5,4,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 2
[1,3,5,6,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2
[1,3,5,6,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2
[1,3,6,2,4,5] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[1,3,6,4,2,5] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2
[1,3,6,4,5,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2
[1,3,6,5,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2
[1,3,6,5,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2
[1,4,2,5,6,3] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 2
[1,4,2,6,3,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[2,3,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[2,3,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,3,6,1,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,3,6,4,1,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,4,1,6,3,5] => [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,4,1,6,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,5,1,3,4,6] => [3,5,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,5,1,4,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,5,3,1,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,5,4,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,6,1,3,5,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,1,4,3,5] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,1,4,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,1,5,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,1,5,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[2,6,3,1,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[2,6,3,1,5,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[3,1,5,2,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[3,1,5,2,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,6,2,4,5] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,6,2,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,6,4,2,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,1,6,5,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[3,2,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
[3,2,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2
[3,2,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001613
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2 - 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2 - 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[2,4,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2 - 1
[3,1,4,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[3,2,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,4,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,2,1,3,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,2,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,3,4,5,6,2] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 4 - 1
[1,3,4,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,4,6,5,2] => [1,6,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 2 - 1
[1,3,5,2,6,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,5,4,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 2 - 1
[1,3,5,6,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 1
[1,3,5,6,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 - 1
[1,3,6,2,4,5] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,6,4,2,5] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 1
[1,3,6,4,5,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 - 1
[1,3,6,5,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 1
[1,3,6,5,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 - 1
[1,4,2,5,6,3] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 2 - 1
[1,4,2,6,3,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[2,3,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[2,3,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,3,6,1,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,3,6,4,1,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,4,1,6,3,5] => [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,4,1,6,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,5,1,3,4,6] => [3,5,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,1,4,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,3,1,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,5,4,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,6,1,3,5,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,4,3,5] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,4,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,5,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,5,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,3,1,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,6,3,1,5,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[3,1,5,2,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[3,1,5,2,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,2,4,5] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,2,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,4,2,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,5,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,2,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[3,2,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[3,2,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001719
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2 - 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2 - 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[2,4,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2 - 1
[3,1,4,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[3,2,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,4,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,2,1,3,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,2,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,3,4,5,6,2] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 4 - 1
[1,3,4,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,4,6,5,2] => [1,6,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 2 - 1
[1,3,5,2,6,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,5,4,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 2 - 1
[1,3,5,6,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 1
[1,3,5,6,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 - 1
[1,3,6,2,4,5] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,6,4,2,5] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 1
[1,3,6,4,5,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 - 1
[1,3,6,5,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 1
[1,3,6,5,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 - 1
[1,4,2,5,6,3] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 2 - 1
[1,4,2,6,3,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[2,3,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[2,3,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,3,6,1,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,3,6,4,1,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,4,1,6,3,5] => [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,4,1,6,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,5,1,3,4,6] => [3,5,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,1,4,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,3,1,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,5,4,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,6,1,3,5,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,4,3,5] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,4,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,5,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,5,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,3,1,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,6,3,1,5,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[3,1,5,2,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[3,1,5,2,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,2,4,5] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,2,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,4,2,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,5,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,2,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[3,2,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[3,2,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001881
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 2 - 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2 - 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2 - 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[2,4,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 2 - 1
[3,1,4,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 1
[3,2,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 2 - 1
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,4,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[3,4,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[4,1,5,2,3] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,1,5,3,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,2,1,3,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 - 1
[4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,2,3,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[4,3,1,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 1
[1,3,4,5,6,2] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 4 - 1
[1,3,4,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,4,6,5,2] => [1,6,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 2 - 1
[1,3,5,2,6,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,5,4,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 2 - 1
[1,3,5,6,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 1
[1,3,5,6,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 - 1
[1,3,6,2,4,5] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[1,3,6,4,2,5] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 1
[1,3,6,4,5,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 - 1
[1,3,6,5,2,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 1
[1,3,6,5,4,2] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 - 1
[1,4,2,5,6,3] => [1,6,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 2 - 1
[1,4,2,6,3,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[2,3,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[2,3,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,3,6,1,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,3,6,4,1,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,4,1,6,3,5] => [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,4,1,6,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,5,1,3,4,6] => [3,5,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,1,4,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,5,3,1,6,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,5,4,1,6,3] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,6,1,3,5,4] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,4,3,5] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,4,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,5,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,1,5,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[2,6,3,1,4,5] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[2,6,3,1,5,4] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[3,1,5,2,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[3,1,5,2,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,2,4,5] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,2,5,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,4,2,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,1,6,5,2,4] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,2,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
[3,2,5,1,4,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 2 - 1
[3,2,5,1,6,4] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000699The toughness times the least common multiple of 1,. St001960The number of descents of a permutation minus one if its first entry is not one. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001863The number of weak excedances of a signed permutation. St001817The number of flag weak exceedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001769The reflection length of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St001645The pebbling number of a connected graph.
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