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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St001464
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 1
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => 3
[1,1,1,0,0,0]
=> [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 5
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 6
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 7
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 6
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St000454
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 43%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [2] => [2] => ([],2)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 - 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3] => [3] => ([],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,5} - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,5} - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,5} - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,5} - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,5} - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,5} - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9} - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5] => ([],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,1,1,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14} - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
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: St000777
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 36%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 36%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,3}
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,3}
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,4,5}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,3,3,3,3,4,5}
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,4,5}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,4,5}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,4,5}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,4,5}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,4,5}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,4,5}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 6
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 6
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001439
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001439: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001439: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> []
=> []
=> [] => ? = 1
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,0,0]
=> []
=> []
=> [] => ? = 2
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 3
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 2
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 5
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 4
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 4
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 4
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 4
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 5
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 4
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 4
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? ∊ {2,3,3,3,4,4,4,4,4,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [11,7,12,4,8,13,2,5,9,14,1,3,6,10,15] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [11,7,12,4,8,13,2,5,9,14,1,3,6,10] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> [10,7,11,4,8,12,2,5,9,1,3,6,13,14] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> [10,7,11,4,8,12,2,5,9,1,3,6,13] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [10,7,11,4,8,12,2,5,9,1,3,6] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> [10,6,11,4,7,2,5,12,13,1,3,8,9,14] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> [10,6,11,4,7,2,5,12,13,1,3,8,9] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8,12,13] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8,12] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10,11,12] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10,11] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> [10,6,3,11,12,2,7,8,13,1,4,5,9,14] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> [10,6,3,11,12,2,7,8,13,1,4,5,9] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5,12,13] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5,12] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> [9,5,3,10,2,6,11,12,1,4,7,8,13] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> [9,5,3,10,2,6,11,12,1,4,7,8] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11,12] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10,11] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => ? ∊ {2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 4
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 4
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 4
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 5
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 4
Description
The number of even weak deficiencies and of odd weak exceedences.
For a permutation $\sigma$, this is the number of indices $i$ such that $\sigma(i) \leq i$ if $i$ is even and $\sigma(i) \geq i$ if $i$ is odd.
According to [1], $\sigma$ is a '''D-permutation''' if all indices have this property and the coefficients of the characteristic polynomial of the homogenized linial arrangement are given by the number of D-permutations with a given number of cycles.
Matching statistic: St001645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1
[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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,5}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,5}
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,5}
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,4,4,5}
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,4,4,5}
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,4,4,5}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(0,1),(0,2),(1,2)],3)
=> 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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[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)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(0,1),(0,2),(1,2)],3)
=> 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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[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)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[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)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(0,1)],2)
=> 2
[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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[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)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
Description
The pebbling number of a connected graph.
Matching statistic: St000456
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 64%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,0,1,0]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 2
[1,1,0,0]
=> [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {2,2,3}
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,3}
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3}
[1,1,1,0,0,0]
=> [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,5}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,5}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,5}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,5}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,5}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,5}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,5}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,5}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,5}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,5] => ([(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,7,7,8,8,9}
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,6] => ([(5,6)],7)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 9
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000771
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 36%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 36%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> ([],2)
=> ? = 2
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,2,3}
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3] => ([],3)
=> ([],3)
=> ? ∊ {2,2,3}
[1,1,1,0,0,0]
=> [3] => ([],3)
=> ([],3)
=> ? ∊ {2,2,3}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,5}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,5}
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,5}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,4,4,4,5}
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ([],4)
=> ? ∊ {2,2,2,3,3,4,4,4,5}
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> ([],4)
=> ? ∊ {2,2,2,3,3,4,4,4,5}
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ([],4)
=> ? ∊ {2,2,2,3,3,4,4,4,5}
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ([],4)
=> ? ∊ {2,2,2,3,3,4,4,4,5}
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ([],4)
=> ? ∊ {2,2,2,3,3,4,4,4,5}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> ([],5)
=> ? ∊ {2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000035
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 43%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,3,2,1,6,5] => 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,5,3,4,2,1] => 2
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 3
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 3
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,5,6,4,3] => ? ∊ {2,2,3,4,4,5}
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [6,5,3,4,2,1,8,7] => ? ∊ {2,2,3,4,4,5}
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => ? ∊ {2,2,3,4,4,5}
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [8,7,3,6,5,4,2,1] => ? ∊ {2,2,3,4,4,5}
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => ? ∊ {2,2,3,4,4,5}
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => ? ∊ {2,2,3,4,4,5}
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,8,6,5] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,6,4,3,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,1,10,7,5,9,4,8,6,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,5,8,7,6,4,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,1,10,7,9,6,4,8,5,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,6,7,5,4,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [4,3,2,1,10,9,7,8,6,5] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [6,5,3,4,2,1,8,7,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [6,5,3,4,2,1,10,9,8,7] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [8,5,3,7,2,6,4,1,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,8,6,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => [10,5,3,9,2,8,7,6,4,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [8,7,3,6,5,4,2,1,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => [10,7,3,6,9,4,2,8,5,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [10,9,3,8,6,5,7,4,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => [10,9,3,8,7,6,5,4,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => 3
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [6,5,4,3,2,1,10,9,8,7] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [8,5,7,4,2,6,3,1,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => [10,5,7,4,2,9,3,8,6,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => [10,5,9,4,2,8,7,6,3,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [8,7,6,4,5,3,2,1,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [10,7,6,4,9,3,2,8,5,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 2
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => [10,7,6,9,5,3,2,8,4,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,4,3,6,5,8,7,12,11,10,9] => 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,4,3,6,5,10,9,8,7,12,11] => 5
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,4,3,6,5,12,11,9,10,8,7] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,4,3,6,5,12,11,10,9,8,7] => 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,4,3,8,7,6,5,10,9,12,11] => 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,4,3,8,7,6,5,12,11,10,9] => 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,4,3,10,9,7,8,6,5,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,4,3,12,9,7,11,6,10,8,5] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,4,3,12,11,7,10,9,8,6,5] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,4,3,10,9,8,7,6,5,12,11] => 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,4,3,12,9,11,8,6,10,7,5] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,4,3,12,11,10,8,9,7,6,5] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,4,3,12,11,10,9,8,7,6,5] => 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,6,5,4,3,8,7,10,9,12,11] => 5
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,6,5,4,3,8,7,12,11,10,9] => 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,6,5,4,3,10,9,8,7,12,11] => 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,6,5,4,3,12,11,9,10,8,7] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,6,5,4,3,12,11,10,9,8,7] => 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,8,7,5,6,4,3,10,9,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,8,7,5,6,4,3,12,11,10,9] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,10,7,5,9,4,8,6,3,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,7,5,9,4,11,6,10,8,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,7,5,11,4,10,9,8,6,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,10,9,5,8,7,6,4,3,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,9,5,8,11,6,4,10,7,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,5,10,8,7,9,6,4,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,5,10,9,8,7,6,4,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> [2,1,6,7,8,5,4,3,10,9,12,11] => [2,1,8,7,6,5,4,3,10,9,12,11] => 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> [2,1,6,7,8,5,4,3,11,12,10,9] => [2,1,8,7,6,5,4,3,12,11,10,9] => 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> [2,1,6,7,9,5,4,10,8,3,12,11] => [2,1,10,7,9,6,4,8,5,3,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,6,7,9,5,4,11,8,12,10,3] => [2,1,12,7,9,6,4,11,5,10,8,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [2,1,7,8,9,10,6,5,4,3,12,11] => [2,1,10,9,8,7,6,5,4,3,12,11] => 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 5
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 4
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000153
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 43%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,3,2,1,6,5] => 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,5,3,4,2,1] => 2
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 3
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 3
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,5,6,4,3] => ? ∊ {2,2,3,4,4,5}
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [6,5,3,4,2,1,8,7] => ? ∊ {2,2,3,4,4,5}
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => ? ∊ {2,2,3,4,4,5}
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [8,7,3,6,5,4,2,1] => ? ∊ {2,2,3,4,4,5}
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => ? ∊ {2,2,3,4,4,5}
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => ? ∊ {2,2,3,4,4,5}
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,8,6,5] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,6,4,3,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,1,10,7,5,9,4,8,6,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,5,8,7,6,4,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,1,10,7,9,6,4,8,5,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,6,7,5,4,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [4,3,2,1,10,9,7,8,6,5] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [6,5,3,4,2,1,8,7,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [6,5,3,4,2,1,10,9,8,7] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [8,5,3,7,2,6,4,1,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,8,6,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => [10,5,3,9,2,8,7,6,4,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [8,7,3,6,5,4,2,1,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => [10,7,3,6,9,4,2,8,5,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [10,9,3,8,6,5,7,4,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => [10,9,3,8,7,6,5,4,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => 3
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [6,5,4,3,2,1,10,9,8,7] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [8,5,7,4,2,6,3,1,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => [10,5,7,4,2,9,3,8,6,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => [10,5,9,4,2,8,7,6,3,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [8,7,6,4,5,3,2,1,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [10,7,6,4,9,3,2,8,5,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 2
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => [10,7,6,9,5,3,2,8,4,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,4,3,6,5,8,7,12,11,10,9] => 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,4,3,6,5,10,9,8,7,12,11] => 5
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,4,3,6,5,12,11,9,10,8,7] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,4,3,6,5,12,11,10,9,8,7] => 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,4,3,8,7,6,5,10,9,12,11] => 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,4,3,8,7,6,5,12,11,10,9] => 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,4,3,10,9,7,8,6,5,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,4,3,12,9,7,11,6,10,8,5] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,4,3,12,11,7,10,9,8,6,5] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,4,3,10,9,8,7,6,5,12,11] => 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,4,3,12,9,11,8,6,10,7,5] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,4,3,12,11,10,8,9,7,6,5] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,4,3,12,11,10,9,8,7,6,5] => 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,6,5,4,3,8,7,10,9,12,11] => 5
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,6,5,4,3,8,7,12,11,10,9] => 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,6,5,4,3,10,9,8,7,12,11] => 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,6,5,4,3,12,11,9,10,8,7] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,6,5,4,3,12,11,10,9,8,7] => 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,8,7,5,6,4,3,10,9,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,8,7,5,6,4,3,12,11,10,9] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,10,7,5,9,4,8,6,3,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,7,5,9,4,11,6,10,8,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,7,5,11,4,10,9,8,6,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,10,9,5,8,7,6,4,3,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,9,5,8,11,6,4,10,7,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,5,10,8,7,9,6,4,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,5,10,9,8,7,6,4,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> [2,1,6,7,8,5,4,3,10,9,12,11] => [2,1,8,7,6,5,4,3,10,9,12,11] => 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> [2,1,6,7,8,5,4,3,11,12,10,9] => [2,1,8,7,6,5,4,3,12,11,10,9] => 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> [2,1,6,7,9,5,4,10,8,3,12,11] => [2,1,10,7,9,6,4,8,5,3,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,6,7,9,5,4,11,8,12,10,3] => [2,1,12,7,9,6,4,11,5,10,8,3] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [2,1,7,8,9,10,6,5,4,3,12,11] => [2,1,10,9,8,7,6,5,4,3,12,11] => 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 5
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 4
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000374
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 43%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,1,6,5] => 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [5,4,1,6,3,2] => 2
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => 3
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => 3
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,7,6,3,8,5,4] => ? ∊ {2,2,3,4,4,5}
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [5,4,1,6,3,2,8,7] => ? ∊ {2,2,3,4,4,5}
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => ? ∊ {2,2,3,4,4,5}
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [7,6,5,4,1,8,3,2] => ? ∊ {2,2,3,4,4,5}
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => ? ∊ {2,2,3,4,4,5}
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => ? ∊ {2,2,3,4,4,5}
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,10,9,8,7] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,9,8,5,10,7,6] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => [2,1,6,5,4,3,10,9,8,7] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,1,7,6,3,8,5,4,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [2,1,9,8,3,7,6,10,5,4] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => [2,1,9,8,7,6,3,10,5,4] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [2,1,9,8,3,10,7,6,5,4] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [2,1,8,7,4,3,10,9,6,5] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [4,3,2,1,9,8,5,10,7,6] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [5,4,1,6,3,2,8,7,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => [5,4,1,6,3,2,10,9,8,7] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [7,6,1,5,4,8,3,2,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [9,8,1,5,4,7,6,10,3,2] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [9,8,7,6,1,5,4,10,3,2] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [7,6,5,4,1,8,3,2,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [9,8,1,7,6,5,4,10,3,2] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [8,7,4,1,9,6,5,10,3,2] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [9,8,7,6,5,4,1,10,3,2] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => 3
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [6,5,4,3,2,1,10,9,8,7] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [7,6,1,8,5,4,3,2,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [9,8,1,7,6,10,5,4,3,2] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [9,8,7,6,1,10,5,4,3,2] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [6,5,2,1,8,7,4,3,10,9] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [9,8,1,6,5,2,10,7,4,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [8,7,2,1,6,5,10,9,4,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => [8,7,6,5,2,1,10,9,4,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 2
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [9,8,1,10,7,6,5,4,3,2] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [8,7,2,1,10,9,6,5,4,3] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [7,6,3,2,1,10,9,8,5,4] => ? ∊ {2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9}
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,12,11,10,9] => [2,1,4,3,6,5,8,7,12,11,10,9] => 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,10,9,8,7,12,11] => [2,1,4,3,6,5,10,9,8,7,12,11] => 5
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,12,9,8,11,10,7] => [2,1,4,3,6,5,11,10,7,12,9,8] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,12,11,10,9,8,7] => [2,1,4,3,6,5,12,11,10,9,8,7] => 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,8,7,6,5,10,9,12,11] => [2,1,4,3,8,7,6,5,10,9,12,11] => 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,8,7,6,5,12,11,10,9] => [2,1,4,3,8,7,6,5,12,11,10,9] => 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,10,7,6,9,8,5,12,11] => [2,1,4,3,9,8,5,10,7,6,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,12,7,6,9,8,11,10,5] => [2,1,4,3,11,10,5,9,8,12,7,6] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,12,7,6,11,10,9,8,5] => [2,1,4,3,11,10,9,8,5,12,7,6] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,10,9,8,7,6,5,12,11] => [2,1,4,3,10,9,8,7,6,5,12,11] => 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,12,9,8,7,6,11,10,5] => [2,1,4,3,11,10,5,12,9,8,7,6] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,12,11,8,7,10,9,6,5] => [2,1,4,3,10,9,6,5,12,11,8,7] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,12,11,10,9,8,7,6,5] => [2,1,4,3,12,11,10,9,8,7,6,5] => 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,6,5,4,3,8,7,10,9,12,11] => [2,1,6,5,4,3,8,7,10,9,12,11] => 5
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,6,5,4,3,8,7,12,11,10,9] => [2,1,6,5,4,3,8,7,12,11,10,9] => 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,6,5,4,3,10,9,8,7,12,11] => [2,1,6,5,4,3,10,9,8,7,12,11] => 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,6,5,4,3,12,9,8,11,10,7] => [2,1,6,5,4,3,11,10,7,12,9,8] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,6,5,4,3,12,11,10,9,8,7] => [2,1,6,5,4,3,12,11,10,9,8,7] => 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,8,5,4,7,6,3,10,9,12,11] => [2,1,7,6,3,8,5,4,10,9,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,8,5,4,7,6,3,12,11,10,9] => [2,1,7,6,3,8,5,4,12,11,10,9] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,10,5,4,7,6,9,8,3,12,11] => [2,1,9,8,3,7,6,10,5,4,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,12,5,4,7,6,9,8,11,10,3] => [2,1,11,10,3,7,6,9,8,12,5,4] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,12,5,4,7,6,11,10,9,8,3] => [2,1,11,10,9,8,3,7,6,12,5,4] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,10,5,4,9,8,7,6,3,12,11] => [2,1,9,8,7,6,3,10,5,4,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,12,5,4,9,8,7,6,11,10,3] => [2,1,11,10,3,9,8,7,6,12,5,4] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,12,5,4,11,8,7,10,9,6,3] => [2,1,10,9,6,3,11,8,7,12,5,4] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [2,1,12,5,4,11,10,9,8,7,6,3] => [2,1,11,10,9,8,7,6,3,12,5,4] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> [2,1,8,7,6,5,4,3,10,9,12,11] => [2,1,8,7,6,5,4,3,10,9,12,11] => 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> [2,1,8,7,6,5,4,3,12,11,10,9] => [2,1,8,7,6,5,4,3,12,11,10,9] => 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> [2,1,10,7,6,5,4,9,8,3,12,11] => [2,1,9,8,3,10,7,6,5,4,12,11] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,12,7,6,5,4,9,8,11,10,3] => [2,1,11,10,3,9,8,12,7,6,5,4] => ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,13,13,13,14,14}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [2,1,10,9,8,7,6,5,4,3,12,11] => [2,1,10,9,8,7,6,5,4,3,12,11] => 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,12,11,10,9,8,7,6,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [4,3,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 5
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [4,3,2,1,6,5,8,7,12,11,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 4
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000189The number of elements in the poset. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000906The length of the shortest maximal chain in a poset. St001176The size of a partition minus its first part. St001279The sum of the parts of an integer partition that are at least two. St001330The hat guessing number of a graph. St001389The number of partitions of the same length below the given integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000455The second largest eigenvalue of a graph if it is integral. St001498The normalised height of a Nakayama algebra with magnitude 1. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000186The sum of the first row in a Gelfand-Tsetlin pattern. 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 $eA$. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001811The Castelnuovo-Mumford regularity of a permutation. St001823The Stasinski-Voll length of a signed permutation. St000422The energy of a graph, if it is integral. St001115The number of even descents of a permutation. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000381The largest part of an integer composition. St000386The number of factors DDU in a Dyck path. St000662The staircase size of the code of a permutation. St000884The number of isolated descents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000703The number of deficiencies of a permutation. St000022The number of fixed points of a permutation. St000390The number of runs of ones in a binary word. St000731The number of double exceedences of a permutation.
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