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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000454
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 0
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001270
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Values
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 0
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,6,5,8,4,3,2] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
Description
The bandwidth of a graph.
The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be
ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$.
We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.
Matching statistic: St000720
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00144: Perfect matchings —rotation⟶ Perfect matchings
St000720: Perfect matchings ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 71%
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00144: Perfect matchings —rotation⟶ Perfect matchings
St000720: Perfect matchings ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [(1,2),(3,4)]
=> [(1,4),(2,3)]
=> 2 = 0 + 2
[1,1,0,0]
=> [(1,4),(2,3)]
=> [(1,3),(2,4)]
=> [(1,3),(2,4)]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> [(1,6),(2,3),(4,5)]
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> [(1,5),(2,3),(4,6)]
=> 2 = 0 + 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> [(1,6),(2,4),(3,5)]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> [(1,5),(2,4),(3,6)]
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> [(1,4),(2,5),(3,6)]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> [(1,7),(2,3),(4,5),(6,8)]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [(1,8),(2,3),(4,6),(5,7)]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [(1,7),(2,3),(4,6),(5,8)]
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> [(1,6),(2,3),(4,7),(5,8)]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> [(1,8),(2,4),(3,5),(6,7)]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [(1,3),(2,4),(5,7),(6,8)]
=> [(1,7),(2,4),(3,5),(6,8)]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> [(1,8),(2,5),(3,6),(4,7)]
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> [(1,7),(2,5),(3,6),(4,8)]
=> 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [(1,4),(2,6),(3,7),(5,8)]
=> [(1,6),(2,5),(3,7),(4,8)]
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10)]
=> [(1,9),(2,3),(4,5),(6,7),(8,10)]
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10)]
=> [(1,10),(2,3),(4,5),(6,8),(7,9)]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10)]
=> [(1,9),(2,3),(4,5),(6,8),(7,10)]
=> 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10)]
=> [(1,8),(2,3),(4,5),(6,9),(7,10)]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [(1,10),(2,3),(4,6),(5,7),(8,9)]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10)]
=> [(1,9),(2,3),(4,6),(5,7),(8,10)]
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10)]
=> [(1,10),(2,3),(4,7),(5,8),(6,9)]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10)]
=> [(1,9),(2,3),(4,7),(5,8),(6,10)]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10)]
=> [(1,8),(2,3),(4,7),(5,9),(6,10)]
=> 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10)]
=> [(1,7),(2,3),(4,8),(5,9),(6,10)]
=> 4 = 2 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [(1,10),(2,4),(3,5),(6,7),(8,9)]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10)]
=> [(1,9),(2,4),(3,5),(6,7),(8,10)]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10)]
=> [(1,10),(2,4),(3,5),(6,8),(7,9)]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10)]
=> [(1,9),(2,4),(3,5),(6,8),(7,10)]
=> 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10)]
=> [(1,8),(2,4),(3,5),(6,9),(7,10)]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [(1,10),(2,5),(3,6),(4,7),(8,9)]
=> 4 = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [(1,4),(2,5),(3,6),(7,9),(8,10)]
=> [(1,9),(2,5),(3,6),(4,7),(8,10)]
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [(1,5),(2,6),(3,7),(4,8),(9,10)]
=> [(1,10),(2,6),(3,7),(4,8),(5,9)]
=> 5 = 3 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [(1,5),(2,6),(3,7),(4,9),(8,10)]
=> [(1,9),(2,6),(3,7),(4,8),(5,10)]
=> 5 = 3 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [(1,5),(2,6),(3,8),(4,9),(7,10)]
=> [(1,8),(2,6),(3,7),(4,9),(5,10)]
=> 5 = 3 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [(1,5),(2,7),(3,8),(4,9),(6,10)]
=> [(1,7),(2,6),(3,8),(4,9),(5,10)]
=> 5 = 3 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [(1,6),(2,7),(3,8),(4,9),(5,10)]
=> [(1,6),(2,7),(3,8),(4,9),(5,10)]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> [(1,11),(2,3),(4,5),(6,7),(8,9),(10,12)]
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> [(1,12),(2,3),(4,5),(6,7),(8,10),(9,11)]
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> [(1,11),(2,3),(4,5),(6,7),(8,10),(9,12)]
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> [(1,10),(2,3),(4,5),(6,7),(8,11),(9,12)]
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> [(1,12),(2,3),(4,5),(6,8),(7,9),(10,11)]
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> [(1,11),(2,3),(4,5),(6,8),(7,9),(10,12)]
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> [(1,12),(2,3),(4,5),(6,9),(7,10),(8,11)]
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> [(1,11),(2,3),(4,5),(6,9),(7,10),(8,12)]
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> [(1,10),(2,3),(4,5),(6,9),(7,11),(8,12)]
=> ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> [(1,9),(2,3),(4,5),(6,10),(7,11),(8,12)]
=> ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> [(1,12),(2,3),(4,6),(5,7),(8,9),(10,11)]
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> [(1,11),(2,3),(4,6),(5,7),(8,9),(10,12)]
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> [(1,12),(2,3),(4,6),(5,7),(8,10),(9,11)]
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> [(1,11),(2,3),(4,6),(5,7),(8,10),(9,12)]
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> [(1,10),(2,3),(4,6),(5,7),(8,11),(9,12)]
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10),(11,12)]
=> [(1,12),(2,3),(4,7),(5,8),(6,9),(10,11)]
=> ? = 2 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> [(1,2),(3,6),(4,7),(5,8),(9,11),(10,12)]
=> [(1,11),(2,3),(4,7),(5,8),(6,9),(10,12)]
=> ? = 2 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12)]
=> [(1,12),(2,3),(4,8),(5,9),(6,10),(7,11)]
=> ? = 3 + 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [(1,2),(3,7),(4,8),(5,9),(6,11),(10,12)]
=> [(1,11),(2,3),(4,8),(5,9),(6,10),(7,12)]
=> ? = 3 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [(1,2),(3,7),(4,8),(5,10),(6,11),(9,12)]
=> [(1,10),(2,3),(4,8),(5,9),(6,11),(7,12)]
=> ? = 3 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [(1,2),(3,7),(4,9),(5,10),(6,11),(8,12)]
=> [(1,9),(2,3),(4,8),(5,10),(6,11),(7,12)]
=> ? = 3 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [(1,2),(3,8),(4,9),(5,10),(6,11),(7,12)]
=> [(1,8),(2,3),(4,9),(5,10),(6,11),(7,12)]
=> ? = 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)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10),(11,12)]
=> [(1,12),(2,4),(3,5),(6,7),(8,9),(10,11)]
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [(1,3),(2,4),(5,6),(7,8),(9,11),(10,12)]
=> [(1,11),(2,4),(3,5),(6,7),(8,9),(10,12)]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10),(11,12)]
=> [(1,12),(2,4),(3,5),(6,7),(8,10),(9,11)]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [(1,3),(2,4),(5,6),(7,9),(8,11),(10,12)]
=> [(1,11),(2,4),(3,5),(6,7),(8,10),(9,12)]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> [(1,3),(2,4),(5,6),(7,10),(8,11),(9,12)]
=> [(1,10),(2,4),(3,5),(6,7),(8,11),(9,12)]
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10),(11,12)]
=> [(1,12),(2,4),(3,5),(6,8),(7,9),(10,11)]
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)]
=> [(1,11),(2,4),(3,5),(6,8),(7,9),(10,12)]
=> ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10),(11,12)]
=> [(1,12),(2,4),(3,5),(6,9),(7,10),(8,11)]
=> ? = 2 + 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [(1,3),(2,4),(5,8),(6,9),(7,11),(10,12)]
=> [(1,11),(2,4),(3,5),(6,9),(7,10),(8,12)]
=> ? = 2 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [(1,3),(2,4),(5,8),(6,10),(7,11),(9,12)]
=> [(1,10),(2,4),(3,5),(6,9),(7,11),(8,12)]
=> ? = 2 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12)]
=> [(1,9),(2,4),(3,5),(6,10),(7,11),(8,12)]
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [(1,3),(2,5),(4,7),(6,9),(8,10),(11,12)]
=> [(1,12),(2,4),(3,6),(5,8),(7,10),(9,11)]
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [(1,3),(2,5),(4,7),(6,9),(8,11),(10,12)]
=> [(1,11),(2,4),(3,6),(5,8),(7,10),(9,12)]
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]
=> [(1,3),(2,5),(4,7),(6,10),(8,11),(9,12)]
=> [(1,10),(2,4),(3,6),(5,8),(7,11),(9,12)]
=> ? = 2 + 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> [(1,4),(2,5),(3,6),(7,8),(9,10),(11,12)]
=> [(1,12),(2,5),(3,6),(4,7),(8,9),(10,11)]
=> ? = 2 + 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> [(1,4),(2,5),(3,6),(7,8),(9,11),(10,12)]
=> [(1,11),(2,5),(3,6),(4,7),(8,9),(10,12)]
=> ? = 2 + 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> [(1,4),(2,5),(3,6),(7,9),(8,10),(11,12)]
=> [(1,12),(2,5),(3,6),(4,7),(8,10),(9,11)]
=> ? = 2 + 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,9),(10,11)]
=> [(1,4),(2,5),(3,6),(7,9),(8,11),(10,12)]
=> [(1,11),(2,5),(3,6),(4,7),(8,10),(9,12)]
=> ? = 2 + 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)]
=> [(1,10),(2,5),(3,6),(4,7),(8,11),(9,12)]
=> ? = 2 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]
=> [(1,4),(2,6),(3,8),(5,9),(7,10),(11,12)]
=> [(1,12),(2,5),(3,7),(4,9),(6,10),(8,11)]
=> ? = 3 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]
=> [(1,4),(2,6),(3,8),(5,9),(7,11),(10,12)]
=> [(1,11),(2,5),(3,7),(4,9),(6,10),(8,12)]
=> ? = 3 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [(1,4),(2,6),(3,8),(5,10),(7,11),(9,12)]
=> [(1,10),(2,5),(3,7),(4,9),(6,11),(8,12)]
=> ? = 3 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,10),(8,9)]
=> [(1,4),(2,6),(3,9),(5,10),(7,11),(8,12)]
=> [(1,9),(2,5),(3,7),(4,10),(6,11),(8,12)]
=> ? = 3 + 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> [(1,5),(2,6),(3,7),(4,8),(9,10),(11,12)]
=> [(1,12),(2,6),(3,7),(4,8),(5,9),(10,11)]
=> ? = 3 + 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> [(1,5),(2,6),(3,7),(4,8),(9,11),(10,12)]
=> [(1,11),(2,6),(3,7),(4,8),(5,9),(10,12)]
=> ? = 3 + 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [(1,6),(2,7),(3,8),(4,9),(5,10),(11,12)]
=> [(1,12),(2,7),(3,8),(4,9),(5,10),(6,11)]
=> ? = 4 + 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]
=> [(1,6),(2,7),(3,8),(4,9),(5,11),(10,12)]
=> [(1,11),(2,7),(3,8),(4,9),(5,10),(6,12)]
=> ? = 4 + 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [(1,12),(2,11),(3,8),(4,7),(5,6),(9,10)]
=> [(1,6),(2,7),(3,8),(4,10),(5,11),(9,12)]
=> [(1,10),(2,7),(3,8),(4,9),(5,11),(6,12)]
=> ? = 4 + 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]
=> [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]
=> 6 = 4 + 2
Description
The size of the largest partition in the oscillating tableau corresponding to the perfect matching.
Equivalently, this is the maximal number of crosses in the corresponding triangular rook filling that can be covered by a rectangle.
Matching statistic: St001207
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 57%
Mp00252: Permutations —restriction⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 57%
Values
[1,0,1,0]
=> [1,2] => [1] => ? = 0
[1,1,0,0]
=> [2,1] => [1] => ? = 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,4] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1] => 3
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => 3
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => 3
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,5,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,5,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,5] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,2,5,4,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,5,4,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => [1,2,5,4,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [1,2,5,4,3] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [1,3,2,4,5] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [1,3,2,5,4] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [1,3,2,5,4] => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => [1,4,3,2,5] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => [1,4,3,2,5] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [1,5,4,3,2] => ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,5,4,3,2] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [1,5,4,3,2] => ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [1,5,4,3,2] => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,5,4,3,2] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [2,1,3,4,5] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [2,1,3,5,4] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => [2,1,3,5,4] => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => [2,1,4,3,5] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => [2,1,5,4,3] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => [2,1,5,4,3] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,4,3] => [2,1,5,4,3] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [2,1,5,4,3] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [2,3,4,5,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,1] => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [2,3,4,5,1] => ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => [3,2,1,4,5] => ? = 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => [3,2,1,4,5] => ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => [3,2,1,5,4] => ? = 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,2,1,5,6,4] => [3,2,1,5,4] => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => [3,2,1,5,4] => ? = 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [3,4,5,2,1] => ? = 3
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,2,6,1] => [3,4,5,2,1] => ? = 3
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => [3,4,5,2,1] => ? = 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,5,2,1] => [3,4,5,2,1] => ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [4,3,2,1,5] => ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [4,3,2,1,5] => ? = 3
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001645
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> ? = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> ? = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([],4)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [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)
=> 5 = 4 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [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)
=> 5 = 4 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,4,3,6,2,1] => [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)
=> 5 = 4 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [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)
=> 5 = 4 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,4,3,2,1] => [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)
=> 5 = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [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)
=> 5 = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 71%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? = 0 + 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,7,3,4,5,6] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 5 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 5 + 1
Description
The 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.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 71%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? = 0 + 2
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? = 0 + 2
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 3 + 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 3 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 3 + 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,7,3,4,5,6] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000291
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 57%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 57%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 1100 => 0100 => 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1010 => 0010 => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 011000 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 001100 => 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 110100 => 010100 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 110010 => 010010 => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 001010 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 01110000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 00111000 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 01011000 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01100010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 00101100 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 01101000 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00110100 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 01010100 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01010010 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 01001010 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 00101010 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 0111100000 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0011110000 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0101110000 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 0111000010 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0010111000 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 0110110000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011011000 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0101011000 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0101100010 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 0110001010 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0010101100 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 0111010000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101000 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0101101000 => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 0110100010 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0010110100 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 0110101000 => ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0011010100 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0101010100 => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0101010010 => ? = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0101001010 => ? = 3 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0100101010 => ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0010101010 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => 011111000000 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => 001111100000 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 110111100000 => 010111100000 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => 011110000010 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => 001011110000 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 111011100000 => 011011100000 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => 001101110000 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 110101110000 => 010101110000 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 110111000010 => 010111000010 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => 011100001010 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 001010111000 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 111101100000 => 011101100000 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => 001110110000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 110110110000 => 010110110000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 111011000010 => 011011000010 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 101011011000 => 001011011000 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 111010110000 => 011010110000 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 101101011000 => 001101011000 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 110101011000 => 010101011000 => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 110101100010 => 010101100010 => ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 110110001010 => 010110001010 => ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => 011000101010 => ? = 3 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 001010101100 => ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 111110100000 => 011110100000 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => 001111010000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 110111010000 => 010111010000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => 011101000010 => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => 001011101000 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 111011010000 => 011011010000 => ? = 1 + 1
Description
The number of descents of a binary word.
Matching statistic: St000292
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 57%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 57%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 1100 => 0100 => 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1010 => 0010 => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 011000 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 001100 => 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 110100 => 010100 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 110010 => 010010 => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 001010 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 01110000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 00111000 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 01011000 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01100010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 00101100 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 01101000 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00110100 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 01010100 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01010010 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 01001010 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 00101010 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 0111100000 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0011110000 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0101110000 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 0111000010 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0010111000 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 0110110000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011011000 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0101011000 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0101100010 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 0110001010 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0010101100 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 0111010000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101000 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0101101000 => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 0110100010 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0010110100 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 0110101000 => ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0011010100 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0101010100 => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0101010010 => ? = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0101001010 => ? = 3 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0100101010 => ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0010101010 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => 011111000000 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => 001111100000 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 110111100000 => 010111100000 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => 011110000010 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => 001011110000 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 111011100000 => 011011100000 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => 001101110000 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 110101110000 => 010101110000 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 110111000010 => 010111000010 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => 011100001010 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 001010111000 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 111101100000 => 011101100000 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => 001110110000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 110110110000 => 010110110000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 111011000010 => 011011000010 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 101011011000 => 001011011000 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 111010110000 => 011010110000 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 101101011000 => 001101011000 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 110101011000 => 010101011000 => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 110101100010 => 010101100010 => ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 110110001010 => 010110001010 => ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => 011000101010 => ? = 3 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 001010101100 => ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 111110100000 => 011110100000 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => 001111010000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 110111010000 => 010111010000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => 011101000010 => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => 001011101000 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 111011010000 => 011011010000 => ? = 1 + 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 57%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 57%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 1100 => 0100 => 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1010 => 0010 => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 011000 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 001100 => 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 110100 => 010100 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 110010 => 010010 => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 001010 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 01110000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 00111000 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 01011000 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01100010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 00101100 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 01101000 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00110100 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 01010100 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01010010 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 01001010 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 00101010 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 0111100000 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0011110000 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0101110000 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 0111000010 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0010111000 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 0110110000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011011000 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0101011000 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0101100010 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 0110001010 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0010101100 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 0111010000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101000 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0101101000 => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 0110100010 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0010110100 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 0110101000 => ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0011010100 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0101010100 => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0101010010 => ? = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0101001010 => ? = 3 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0100101010 => ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0010101010 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => 011111000000 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => 001111100000 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 110111100000 => 010111100000 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => 011110000010 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => 001011110000 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 111011100000 => 011011100000 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => 001101110000 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 110101110000 => 010101110000 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 110111000010 => 010111000010 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => 011100001010 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 001010111000 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 111101100000 => 011101100000 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => 001110110000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 110110110000 => 010110110000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 111011000010 => 011011000010 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 101011011000 => 001011011000 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 111010110000 => 011010110000 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 101101011000 => 001101011000 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 110101011000 => 010101011000 => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 110101100010 => 010101100010 => ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 110110001010 => 010110001010 => ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => 011000101010 => ? = 3 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 001010101100 => ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 111110100000 => 011110100000 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => 001111010000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 110111010000 => 010111010000 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => 011101000010 => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => 001011101000 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 111011010000 => 011011010000 => ? = 1 + 1
Description
The number of runs of ones in a binary word.
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