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Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St001315
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(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => ([],2)
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 3
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 4
Description
The dissociation number of a graph.
Matching statistic: St000837
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 86%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 1 = 2 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,3,5,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,5,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [1,5,2,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [1,2,4,5,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,5,2,3,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [1,2,5,3,4] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [1,2,3,5,4] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [1,6,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [1,5,6,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => [1,6,4,5,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,4,1] => [1,4,6,5,3,2] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [1,4,5,6,3,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => [1,6,5,3,4,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => [1,5,6,3,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,3,6,1] => [1,6,3,5,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,3,1] => [1,3,6,5,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => [1,3,5,6,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => [1,6,3,4,5,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,4,6,3,1] => [1,3,6,4,5,2] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,4,3,1] => [1,3,4,6,5,2] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [1,3,4,5,6,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [1,6,5,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => [1,5,6,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => [1,6,4,5,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,5,6,4,1] => [1,4,6,5,2,3] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => [1,4,5,6,2,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,2,5,6,1] => [1,6,5,2,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,2,6,5,1] => [1,5,6,2,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,2,6,1] => [1,6,2,5,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => [1,2,6,5,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,5,2,1] => [1,2,5,6,4,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,4,2,6,1] => [1,6,2,4,5,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,4,6,2,1] => [1,2,6,4,5,3] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,4,2,1] => [1,2,4,6,5,3] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => [1,2,4,5,6,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => [1,6,7,5,4,3,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,5,7,1] => [1,7,5,6,4,3,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,7,5,1] => [1,5,7,6,4,3,2] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => [1,5,6,7,4,3,2] => ? = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => [1,7,6,4,5,3,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => [1,6,7,4,5,3,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,6,4,7,1] => [1,7,4,6,5,3,2] => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,5,6,7,4,1] => [1,4,7,6,5,3,2] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,6,4,1] => [1,4,6,7,5,3,2] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => [1,7,4,5,6,3,2] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => [1,7,6,5,3,4,2] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => [1,6,7,5,3,4,2] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => [1,7,5,6,3,4,2] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,6,7,5,1] => [1,5,7,6,3,4,2] => ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => [1,5,6,7,3,4,2] => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,5,3,6,7,1] => [1,7,6,3,5,4,2] => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [2,4,5,3,7,6,1] => [1,6,7,3,5,4,2] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,3,7,1] => [1,7,3,6,5,4,2] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,5,3,7,1] => [1,7,3,5,6,4,2] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => [1,7,6,3,4,5,2] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,5,4,3,7,6,1] => [1,6,7,3,4,5,2] => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,4,6,3,7,1] => [1,7,3,6,4,5,2] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,4,3,7,1] => [1,7,3,4,6,5,2] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => [1,7,3,4,5,6,2] => ? = 4 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => [1,7,6,5,4,2,3] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,7,6,1] => [1,6,7,5,4,2,3] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => [1,7,5,6,4,2,3] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,2,4,6,7,5,1] => [1,5,7,6,4,2,3] => ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => [1,5,6,7,4,2,3] => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,7,6,1] => [1,6,7,4,5,2,3] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [3,2,5,6,4,7,1] => [1,7,4,6,5,2,3] => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => [1,7,4,5,6,2,3] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,1] => [1,7,6,5,2,4,3] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [3,4,2,5,7,6,1] => [1,6,7,5,2,4,3] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [3,4,2,6,5,7,1] => [1,7,5,6,2,4,3] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [3,4,2,6,7,5,1] => [1,5,7,6,2,4,3] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,4,2,7,6,5,1] => [1,5,6,7,2,4,3] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => [1,7,6,2,5,4,3] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [3,4,5,2,7,6,1] => [1,6,7,2,5,4,3] => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,2,7,1] => [1,7,2,6,5,4,3] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,5,2,7,1] => [1,7,2,5,6,4,3] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => [1,7,6,2,4,5,3] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [3,5,4,2,7,6,1] => [1,6,7,2,4,5,3] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [3,5,4,6,2,7,1] => [1,7,2,6,4,5,3] => ? = 3 - 1
Description
The number of ascents of distance 2 of a permutation.
This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Matching statistic: St001526
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St001526: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 3
[1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5
[1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 3
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000836
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,4,1] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,3,6,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,3,1] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,4,6,3,1] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,4,3,1] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,5,6,4,1] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,2,5,6,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,2,6,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,2,6,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,5,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,4,2,6,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,4,6,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,4,2,1] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,5,7,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,7,5,1] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => ? = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,6,4,7,1] => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,5,6,7,4,1] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,6,4,1] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,6,5,7,4,1] => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,6,7,5,4,1] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,6,7,5,1] => ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,5,3,6,7,1] => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [2,4,5,3,7,6,1] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,3,7,1] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,3,1] => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,6,3,1] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,5,3,7,1] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,5,7,3,1] => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,5,3,1] => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,6,5,3,1] => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,5,4,3,7,6,1] => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,4,6,3,7,1] => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,5,4,6,7,3,1] => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,5,4,7,6,3,1] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,4,3,7,1] => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,5,6,4,7,3,1] => ? = 4 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,5,6,7,4,3,1] => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,6,4,3,1] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,6,5,4,7,3,1] => ? = 4 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,6,5,7,4,3,1] => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,5,4,3,1] => ? = 5 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 5 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,7,6,1] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,2,4,6,7,5,1] => ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,7,6,1] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [3,2,5,6,4,7,1] => ? = 3 - 1
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St000628
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> 10 => 11 => 00 => 0 = 1 - 1
[1,0,1,0]
=> 1010 => 1101 => 0010 => 1 = 2 - 1
[1,1,0,0]
=> 1100 => 1101 => 0010 => 1 = 2 - 1
[1,0,1,0,1,0]
=> 101010 => 110101 => 001010 => 1 = 2 - 1
[1,0,1,1,0,0]
=> 101100 => 110101 => 001010 => 1 = 2 - 1
[1,1,0,0,1,0]
=> 110010 => 110101 => 001010 => 1 = 2 - 1
[1,1,0,1,0,0]
=> 110100 => 111001 => 000110 => 2 = 3 - 1
[1,1,1,0,0,0]
=> 111000 => 111001 => 000110 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11010101 => 00101010 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11010101 => 00101010 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11010101 => 00101010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11011001 => 00100110 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11011001 => 00100110 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 11010101 => 00101010 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 11010101 => 00101010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 11100101 => 00011010 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 11101001 => 00010110 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 11101001 => 00010110 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 11100101 => 00011010 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 11101001 => 00010110 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 11110001 => 00001110 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 11110001 => 00001110 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1101010101 => 0010101010 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 1101010101 => 0010101010 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1101010101 => 0010101010 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 1101011001 => 0010100110 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 1101011001 => 0010100110 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1101010101 => 0010101010 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 1101010101 => 0010101010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 1101100101 => 0010011010 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 1101101001 => 0010010110 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 1101101001 => 0010010110 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1101100101 => 0010011010 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 1101101001 => 0010010110 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 1101110001 => 0010001110 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 1101110001 => 0010001110 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1101010101 => 0010101010 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1101010101 => 0010101010 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1101010101 => 0010101010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1101011001 => 0010100110 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1101011001 => 0010100110 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1110010101 => 0001101010 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 1110010101 => 0001101010 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1110100101 => 0001011010 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1110101001 => 0001010110 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 1110101001 => 0001010110 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1110100101 => 0001011010 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1110101001 => 0001010110 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1110110001 => 0001001110 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 1110110001 => 0001001110 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 110101010101 => 001010101010 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 110101010101 => 001010101010 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => 110101010101 => 001010101010 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => 110101011001 => 001010100110 => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 110101011001 => 001010100110 => ? = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => 110101010101 => 001010101010 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => 110101010101 => 001010101010 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 101011010010 => 110101100101 => 001010011010 => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => 110101101001 => 001010010110 => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> 101011011000 => 110101101001 => 001010010110 => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> 101011100010 => 110101100101 => 001010011010 => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => 110101101001 => 001010010110 => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => 110101110001 => 001010001110 => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => 110101110001 => 001010001110 => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 101100101010 => 110101010101 => 001010101010 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 101100101100 => 110101010101 => 001010101010 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 101100110010 => 110101010101 => 001010101010 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 101100110100 => 110101011001 => 001010100110 => ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> 101100111000 => 110101011001 => 001010100110 => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 101101001010 => 110110010101 => 001001101010 => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 101101001100 => 110110010101 => 001001101010 => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 101101010010 => 110110100101 => 001001011010 => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => 110110101001 => 001001010110 => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> 101101011000 => 110110101001 => 001001010110 => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> 101101100010 => 110110100101 => 001001011010 => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> 101101100100 => 110110101001 => 001001010110 => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> 101101101000 => 110110110001 => 001001001110 => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => 110110110001 => 001001001110 => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> 101110001010 => 110110010101 => 001001101010 => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> 101110001100 => 110110010101 => 001001101010 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> 101110010010 => 110110100101 => 001001011010 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => 110110101001 => 001001010110 => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> 101110011000 => 110110101001 => 001001010110 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> 101110100010 => 110111000101 => 001000111010 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> 101110100100 => 110111001001 => 001000110110 => ? = 4 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> 101110101000 => 110111010001 => 001000101110 => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => 110111010001 => 001000101110 => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => 110111000101 => 001000111010 => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => 110111001001 => 001000110110 => ? = 4 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> 101111001000 => 110111010001 => 001000101110 => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => 110111100001 => 001000011110 => ? = 5 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => 110111100001 => 001000011110 => ? = 5 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => 110101010101 => 001010101010 => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 110010101100 => 110101010101 => 001010101010 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 110010110010 => 110101010101 => 001010101010 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 110010110100 => 110101011001 => 001010100110 => ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> 110010111000 => 110101011001 => 001010100110 => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 110011001010 => 110101010101 => 001010101010 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> 110011001100 => 110101010101 => 001010101010 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 110011010010 => 110101100101 => 001010011010 => ? = 3 - 1
Description
The balance of a binary word.
The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1].
A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Matching statistic: St001960
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 57%
Mp00064: Permutations —reverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1] => [1] => [1] => ? = 1 - 2
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 0 = 2 - 2
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 0 = 2 - 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0 = 2 - 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => [3,1,2] => 0 = 2 - 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1,3] => [2,1,3] => 0 = 2 - 2
[1,1,0,1,0,0]
=> [2,1,3] => [3,1,2] => [1,3,2] => 1 = 3 - 2
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 1 = 3 - 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,4,3] => [4,1,2,3] => 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => [3,1,2,4] => 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => [4,3,1,2] => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [2,4,1,3] => 0 = 2 - 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,2,4] => [1,3,2,4] => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,1,2,3] => [1,2,4,3] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4,2,1,3] => [2,1,4,3] => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => [1,4,3,2] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,4,3,5] => [4,1,2,3,5] => 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,5,2,3,4] => 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => [5,4,1,2,3] => 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,3,2,4,5] => [3,1,2,4,5] => 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,3,2,5,4] => [3,5,1,2,4] => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,2,5,3,4] => 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,5,2,4,3] => [5,1,4,2,3] => 1 = 3 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => [4,3,1,2,5] => 1 = 3 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,5,3,2,4] => [5,1,3,2,4] => 1 = 3 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,2,3] => 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [5,4,3,1,2] => 2 = 4 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,5,1,3,4] => 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,4,1,3,5] => 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,1,5,3,4] => [5,2,1,3,4] => 1 = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => [5,2,4,1,3] => 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,2,4,5] => [1,3,2,4,5] => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,2,5,4] => [1,3,5,2,4] => 1 = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,3,5] => [1,2,4,3,5] => 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,5,4] => 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [5,1,2,4,3] => [5,1,2,4,3] => 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2,5] => 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [5,1,3,2,4] => [3,1,2,5,4] => 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [5,1,4,2,3] => [1,5,2,4,3] => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [5,4,1,3,2] => 2 = 4 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => ? = 3 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => ? = 3 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,2,4,3,6,5] => [4,6,1,2,3,5] => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => ? = 3 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 3 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ? = 3 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,2,5,4,3,6] => [5,4,1,2,3,6] => ? = 3 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [1,2,6,4,3,5] => [6,1,4,2,3,5] => ? = 3 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [1,2,6,5,3,4] => [1,6,5,2,3,4] => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ? = 4 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [1,3,2,4,6,5] => [3,6,1,2,4,5] => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => ? = 3 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [1,3,2,6,5,4] => [6,3,5,1,2,4] => ? = 3 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => ? = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [1,4,2,3,6,5] => [1,4,6,2,3,5] => ? = 3 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [1,5,2,3,4,6] => [1,2,5,3,4,6] => ? = 3 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [1,6,2,3,4,5] => [1,2,3,6,4,5] => ? = 3 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => [1,6,2,3,5,4] => [6,1,2,5,3,4] => ? = 3 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [1,5,2,4,3,6] => [5,1,4,2,3,6] => ? = 3 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [1,6,2,4,3,5] => [6,1,2,4,3,5] => ? = 3 - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => ? = 4 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [1,6,2,5,4,3] => [6,5,1,4,2,3] => ? = 4 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,4,3,2,5,6] => [4,3,1,2,5,6] => ? = 3 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [1,4,3,2,6,5] => [4,3,6,1,2,5] => ? = 3 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => [1,5,3,2,4,6] => [5,1,3,2,4,6] => ? = 3 - 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => [1,6,3,2,4,5] => [3,1,2,6,4,5] => ? = 3 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [1,6,3,2,5,4] => [1,6,3,5,2,4] => ? = 3 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [1,5,4,2,3,6] => [1,5,4,2,3,6] => ? = 4 - 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [1,6,4,2,3,5] => [1,6,2,4,3,5] => ? = 4 - 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [1,6,5,2,3,4] => [1,2,6,5,3,4] => ? = 4 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [1,6,5,2,4,3] => [6,1,5,4,2,3] => ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,5,4,3,2,6] => [5,4,3,1,2,6] => ? = 4 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [1,6,4,3,2,5] => [6,4,1,3,2,5] => ? = 4 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [1,6,5,3,2,4] => [6,1,5,3,2,4] => ? = 4 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [1,6,5,4,2,3] => [1,6,5,4,2,3] => ? = 5 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,6,5,4,3,2] => [6,5,4,3,1,2] => ? = 5 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,6,1,3,4,5] => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [2,1,3,5,4,6] => [2,5,1,3,4,6] => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [2,1,3,6,4,5] => [6,2,1,3,4,5] => ? = 3 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => [2,1,3,6,5,4] => [6,2,5,1,3,4] => ? = 3 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => [2,1,4,3,5,6] => [2,4,1,3,5,6] => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => [2,4,6,1,3,5] => ? = 2 - 2
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001118
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1,0]
=> ([],1)
=> ([],1)
=> ? = 1 - 2
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> ? = 2 - 2
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> ? = 2 - 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 2
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 2
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 2
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 3 - 2
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 3 - 2
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 2
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 3 - 2
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 3 - 2
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 2
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4 - 2
[1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 4 - 2
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 4 - 2
[1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 5 - 2
[1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 3 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 3 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 3 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 4 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 3 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1 = 3 - 2
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1,0]
=> ([],1)
=> ([],1)
=> ? = 1 - 3
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> ? = 2 - 3
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> ? = 2 - 3
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 3
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 3
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 3
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 3 - 3
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0 = 3 - 3
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 3
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 3
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 3
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 3 - 3
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 3 - 3
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 3
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 3
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 3 - 3
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 3 - 3
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 0 = 3 - 3
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 0 = 3 - 3
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 4 - 3
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ? = 4 - 3
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 3
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 3
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 3
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 3
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 3
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 3
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 3
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 3
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 3
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 3
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 3
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 3
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 3
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 3
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 3
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 3
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 3
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 3 - 3
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 3
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 3
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 0 = 3 - 3
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 3
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 3
[1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4 - 3
[1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 4 - 3
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 3
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 3
[1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 4 - 3
[1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 5 - 3
[1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 2 - 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 3 - 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,1,0,0,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
[1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 0 = 3 - 3
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000793
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [(1,2)]
=> {{1,2}}
=> {{1,2}}
=> 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 2
[1,1,0,0]
=> [(1,4),(2,3)]
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> {{1,2},{3,4},{5,6}}
=> 2
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> {{1,2},{3,5},{4,6}}
=> 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> {{1,3},{2,4},{5,6}}
=> 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> {{1,3},{2,5},{4,6}}
=> 3
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> {{1,4},{2,5},{3,6}}
=> 3
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6},{7,8}}
=> 2
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> {{1,2},{3,4},{5,8},{6,7}}
=> {{1,2},{3,4},{5,7},{6,8}}
=> 2
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> {{1,2},{3,5},{4,6},{7,8}}
=> 2
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> {{1,2},{3,5},{4,7},{6,8}}
=> 3
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> {{1,2},{3,8},{4,7},{5,6}}
=> {{1,2},{3,6},{4,7},{5,8}}
=> 3
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> {{1,3},{2,4},{5,6},{7,8}}
=> 2
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> {{1,4},{2,3},{5,8},{6,7}}
=> {{1,3},{2,4},{5,7},{6,8}}
=> 2
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> {{1,3},{2,5},{4,6},{7,8}}
=> 3
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> {{1,3},{2,5},{4,7},{6,8}}
=> 3
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> {{1,8},{2,3},{4,7},{5,6}}
=> {{1,3},{2,6},{4,7},{5,8}}
=> 3
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> {{1,4},{2,5},{3,6},{7,8}}
=> 3
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> {{1,4},{2,5},{3,7},{6,8}}
=> 3
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> {{1,8},{2,7},{3,4},{5,6}}
=> {{1,4},{2,6},{3,7},{5,8}}
=> 4
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9}}
=> {{1,2},{3,4},{5,6},{7,9},{8,10}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> {{1,2},{3,4},{5,8},{6,7},{9,10}}
=> {{1,2},{3,4},{5,7},{6,8},{9,10}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9}}
=> {{1,2},{3,4},{5,7},{6,9},{8,10}}
=> ? = 3
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> {{1,2},{3,4},{5,10},{6,9},{7,8}}
=> {{1,2},{3,4},{5,8},{6,9},{7,10}}
=> ? = 3
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> {{1,2},{3,6},{4,5},{7,8},{9,10}}
=> {{1,2},{3,5},{4,6},{7,8},{9,10}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> {{1,2},{3,6},{4,5},{7,10},{8,9}}
=> {{1,2},{3,5},{4,6},{7,9},{8,10}}
=> ? = 2
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> {{1,2},{3,8},{4,5},{6,7},{9,10}}
=> {{1,2},{3,5},{4,7},{6,8},{9,10}}
=> ? = 3
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> {{1,2},{3,5},{4,7},{6,9},{8,10}}
=> ? = 3
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> {{1,2},{3,10},{4,5},{6,9},{7,8}}
=> {{1,2},{3,5},{4,8},{6,9},{7,10}}
=> ? = 3
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> {{1,2},{3,8},{4,7},{5,6},{9,10}}
=> {{1,2},{3,6},{4,7},{5,8},{9,10}}
=> ? = 3
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> {{1,2},{3,10},{4,7},{5,6},{8,9}}
=> {{1,2},{3,6},{4,7},{5,9},{8,10}}
=> ? = 3
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> {{1,2},{3,10},{4,9},{5,6},{7,8}}
=> {{1,2},{3,6},{4,8},{5,9},{7,10}}
=> ? = 4
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> {{1,2},{3,10},{4,9},{5,8},{6,7}}
=> {{1,2},{3,7},{4,8},{5,9},{6,10}}
=> ? = 4
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> {{1,4},{2,3},{5,6},{7,8},{9,10}}
=> {{1,3},{2,4},{5,6},{7,8},{9,10}}
=> ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> {{1,4},{2,3},{5,6},{7,10},{8,9}}
=> {{1,3},{2,4},{5,6},{7,9},{8,10}}
=> ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> {{1,4},{2,3},{5,8},{6,7},{9,10}}
=> {{1,3},{2,4},{5,7},{6,8},{9,10}}
=> ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> {{1,4},{2,3},{5,10},{6,7},{8,9}}
=> {{1,3},{2,4},{5,7},{6,9},{8,10}}
=> ? = 3
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> {{1,4},{2,3},{5,10},{6,9},{7,8}}
=> {{1,3},{2,4},{5,8},{6,9},{7,10}}
=> ? = 3
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> {{1,6},{2,3},{4,5},{7,8},{9,10}}
=> {{1,3},{2,5},{4,6},{7,8},{9,10}}
=> ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> {{1,6},{2,3},{4,5},{7,10},{8,9}}
=> {{1,3},{2,5},{4,6},{7,9},{8,10}}
=> ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> {{1,8},{2,3},{4,5},{6,7},{9,10}}
=> {{1,3},{2,5},{4,7},{6,8},{9,10}}
=> ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> {{1,3},{2,5},{4,7},{6,9},{8,10}}
=> ? = 3
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> {{1,10},{2,3},{4,5},{6,9},{7,8}}
=> {{1,3},{2,5},{4,8},{6,9},{7,10}}
=> ? = 3
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> {{1,8},{2,3},{4,7},{5,6},{9,10}}
=> {{1,3},{2,6},{4,7},{5,8},{9,10}}
=> ? = 3
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> {{1,10},{2,3},{4,7},{5,6},{8,9}}
=> {{1,3},{2,6},{4,7},{5,9},{8,10}}
=> ? = 3
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> {{1,10},{2,3},{4,9},{5,6},{7,8}}
=> {{1,3},{2,6},{4,8},{5,9},{7,10}}
=> ? = 4
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> {{1,10},{2,3},{4,9},{5,8},{6,7}}
=> {{1,3},{2,7},{4,8},{5,9},{6,10}}
=> ? = 4
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> {{1,6},{2,5},{3,4},{7,8},{9,10}}
=> {{1,4},{2,5},{3,6},{7,8},{9,10}}
=> ? = 3
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> {{1,6},{2,5},{3,4},{7,10},{8,9}}
=> {{1,4},{2,5},{3,6},{7,9},{8,10}}
=> ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> {{1,8},{2,5},{3,4},{6,7},{9,10}}
=> {{1,4},{2,5},{3,7},{6,8},{9,10}}
=> ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> {{1,10},{2,5},{3,4},{6,7},{8,9}}
=> {{1,4},{2,5},{3,7},{6,9},{8,10}}
=> ? = 3
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> {{1,10},{2,5},{3,4},{6,9},{7,8}}
=> {{1,4},{2,5},{3,8},{6,9},{7,10}}
=> ? = 3
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> {{1,8},{2,7},{3,4},{5,6},{9,10}}
=> {{1,4},{2,6},{3,7},{5,8},{9,10}}
=> ? = 4
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> {{1,10},{2,7},{3,4},{5,6},{8,9}}
=> {{1,4},{2,6},{3,7},{5,9},{8,10}}
=> ? = 4
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> {{1,10},{2,9},{3,4},{5,6},{7,8}}
=> {{1,4},{2,6},{3,8},{5,9},{7,10}}
=> ? = 4
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> {{1,10},{2,9},{3,4},{5,8},{6,7}}
=> {{1,4},{2,7},{3,8},{5,9},{6,10}}
=> ? = 4
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> {{1,8},{2,7},{3,6},{4,5},{9,10}}
=> {{1,5},{2,6},{3,7},{4,8},{9,10}}
=> ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> {{1,10},{2,7},{3,6},{4,5},{8,9}}
=> {{1,5},{2,6},{3,7},{4,9},{8,10}}
=> ? = 4
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> {{1,10},{2,9},{3,6},{4,5},{7,8}}
=> {{1,5},{2,6},{3,8},{4,9},{7,10}}
=> ? = 4
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> {{1,10},{2,9},{3,8},{4,5},{6,7}}
=> {{1,5},{2,7},{3,8},{4,9},{6,10}}
=> ? = 5
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> ? = 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> {{1,2},{3,4},{5,6},{7,8},{9,12},{10,11}}
=> {{1,2},{3,4},{5,6},{7,8},{9,11},{10,12}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9},{11,12}}
=> {{1,2},{3,4},{5,6},{7,9},{8,10},{11,12}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> {{1,2},{3,4},{5,6},{7,12},{8,9},{10,11}}
=> {{1,2},{3,4},{5,6},{7,9},{8,11},{10,12}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> {{1,2},{3,4},{5,6},{7,12},{8,11},{9,10}}
=> {{1,2},{3,4},{5,6},{7,10},{8,11},{9,12}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> {{1,2},{3,4},{5,8},{6,7},{9,10},{11,12}}
=> {{1,2},{3,4},{5,7},{6,8},{9,10},{11,12}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> {{1,2},{3,4},{5,8},{6,7},{9,12},{10,11}}
=> {{1,2},{3,4},{5,7},{6,8},{9,11},{10,12}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9},{11,12}}
=> {{1,2},{3,4},{5,7},{6,9},{8,10},{11,12}}
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> {{1,2},{3,4},{5,12},{6,7},{8,9},{10,11}}
=> {{1,2},{3,4},{5,7},{6,9},{8,11},{10,12}}
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> {{1,2},{3,4},{5,12},{6,7},{8,11},{9,10}}
=> {{1,2},{3,4},{5,7},{6,10},{8,11},{9,12}}
=> ? = 3
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition.
To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$.
Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row.
Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$.
This statistic is the length of the longest partition on the diagonal of the diagram.
Matching statistic: St000381
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> 10 => 11 => [1,1,1] => 1
[1,0,1,0]
=> 1010 => 1101 => [1,1,2,1] => 2
[1,1,0,0]
=> 1100 => 1101 => [1,1,2,1] => 2
[1,0,1,0,1,0]
=> 101010 => 110101 => [1,1,2,2,1] => 2
[1,0,1,1,0,0]
=> 101100 => 110101 => [1,1,2,2,1] => 2
[1,1,0,0,1,0]
=> 110010 => 110101 => [1,1,2,2,1] => 2
[1,1,0,1,0,0]
=> 110100 => 111001 => [1,1,1,3,1] => 3
[1,1,1,0,0,0]
=> 111000 => 111001 => [1,1,1,3,1] => 3
[1,0,1,0,1,0,1,0]
=> 10101010 => 11010101 => [1,1,2,2,2,1] => 2
[1,0,1,0,1,1,0,0]
=> 10101100 => 11010101 => [1,1,2,2,2,1] => 2
[1,0,1,1,0,0,1,0]
=> 10110010 => 11010101 => [1,1,2,2,2,1] => 2
[1,0,1,1,0,1,0,0]
=> 10110100 => 11011001 => [1,1,2,1,3,1] => 3
[1,0,1,1,1,0,0,0]
=> 10111000 => 11011001 => [1,1,2,1,3,1] => 3
[1,1,0,0,1,0,1,0]
=> 11001010 => 11010101 => [1,1,2,2,2,1] => 2
[1,1,0,0,1,1,0,0]
=> 11001100 => 11010101 => [1,1,2,2,2,1] => 2
[1,1,0,1,0,0,1,0]
=> 11010010 => 11100101 => [1,1,1,3,2,1] => 3
[1,1,0,1,0,1,0,0]
=> 11010100 => 11101001 => [1,1,1,2,3,1] => 3
[1,1,0,1,1,0,0,0]
=> 11011000 => 11101001 => [1,1,1,2,3,1] => 3
[1,1,1,0,0,0,1,0]
=> 11100010 => 11100101 => [1,1,1,3,2,1] => 3
[1,1,1,0,0,1,0,0]
=> 11100100 => 11101001 => [1,1,1,2,3,1] => 3
[1,1,1,0,1,0,0,0]
=> 11101000 => 11110001 => [1,1,1,1,4,1] => 4
[1,1,1,1,0,0,0,0]
=> 11110000 => 11110001 => [1,1,1,1,4,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1101010101 => [1,1,2,2,2,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 1101010101 => [1,1,2,2,2,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1101010101 => [1,1,2,2,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 1101011001 => [1,1,2,2,1,3,1] => ? = 3
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 1101011001 => [1,1,2,2,1,3,1] => ? = 3
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1101010101 => [1,1,2,2,2,2,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 1101010101 => [1,1,2,2,2,2,1] => ? = 2
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 1101100101 => [1,1,2,1,3,2,1] => ? = 3
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 1101101001 => [1,1,2,1,2,3,1] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 1101101001 => [1,1,2,1,2,3,1] => ? = 3
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1101100101 => [1,1,2,1,3,2,1] => ? = 3
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 1101101001 => [1,1,2,1,2,3,1] => ? = 3
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 1101110001 => [1,1,2,1,1,4,1] => ? = 4
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 1101110001 => [1,1,2,1,1,4,1] => ? = 4
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1101010101 => [1,1,2,2,2,2,1] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1101010101 => [1,1,2,2,2,2,1] => ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1101010101 => [1,1,2,2,2,2,1] => ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1101011001 => [1,1,2,2,1,3,1] => ? = 3
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1101011001 => [1,1,2,2,1,3,1] => ? = 3
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1110010101 => [1,1,1,3,2,2,1] => ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 1110010101 => [1,1,1,3,2,2,1] => ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1110100101 => [1,1,1,2,3,2,1] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1110101001 => [1,1,1,2,2,3,1] => ? = 3
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 1110101001 => [1,1,1,2,2,3,1] => ? = 3
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1110100101 => [1,1,1,2,3,2,1] => ? = 3
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1110101001 => [1,1,1,2,2,3,1] => ? = 3
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1110110001 => [1,1,1,2,1,4,1] => ? = 4
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 1110110001 => [1,1,1,2,1,4,1] => ? = 4
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1110010101 => [1,1,1,3,2,2,1] => ? = 3
[1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 1110010101 => [1,1,1,3,2,2,1] => ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 1110100101 => [1,1,1,2,3,2,1] => ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 1110101001 => [1,1,1,2,2,3,1] => ? = 3
[1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 1110101001 => [1,1,1,2,2,3,1] => ? = 3
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 1111000101 => [1,1,1,1,4,2,1] => ? = 4
[1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 1111001001 => [1,1,1,1,3,3,1] => ? = 4
[1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 1111010001 => [1,1,1,1,2,4,1] => ? = 4
[1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 1111010001 => [1,1,1,1,2,4,1] => ? = 4
[1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 1111000101 => [1,1,1,1,4,2,1] => ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 1111001001 => [1,1,1,1,3,3,1] => ? = 4
[1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 1111010001 => [1,1,1,1,2,4,1] => ? = 4
[1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 1111100001 => [1,1,1,1,1,5,1] => ? = 5
[1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 1111100001 => [1,1,1,1,1,5,1] => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 110101010101 => [1,1,2,2,2,2,2,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 110101010101 => [1,1,2,2,2,2,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => 110101010101 => [1,1,2,2,2,2,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => 110101011001 => [1,1,2,2,2,1,3,1] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 110101011001 => [1,1,2,2,2,1,3,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => 110101010101 => [1,1,2,2,2,2,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => 110101010101 => [1,1,2,2,2,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 101011010010 => 110101100101 => [1,1,2,2,1,3,2,1] => ? = 3
Description
The largest part of an integer composition.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
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