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Your data matches 20 different statistics following compositions of up to 3 maps.
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Matching statistic: St000639
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2,1] => ([],2)
=> 2
[1,1,0,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 3
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 4
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 4
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 5
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 5
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 5
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 5
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 6
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 6
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 5
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 6
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 7
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 7
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 6
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 7
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 8
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 8
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 7
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> 8
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 8
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 8
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 8
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 9
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 10
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 9
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 7
Description
The number of relations in a poset.
This is the number of intervals $x,y$ with $x\leq y$ in the poset, and therefore the dimension of the posets incidence algebra.
Matching statistic: St000641
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000641: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000641: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2,1] => ([],2)
=> 2
[1,1,0,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 3
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 4
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 4
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 5
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 5
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 5
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 5
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 6
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 6
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 5
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 6
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 7
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 7
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 6
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 7
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 8
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 8
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 7
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> 8
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 8
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 8
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 8
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 9
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 10
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 9
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 7
Description
The number of non-empty boolean intervals in a poset.
Matching statistic: St001034
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 6 = 5 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 9 = 8 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 9 = 8 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 10 = 9 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 10 = 9 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 9 = 8 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 10 = 9 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 11 = 10 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 10 = 9 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 9 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 10 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 9 + 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 10 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 9 + 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 10 + 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 11 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 12 + 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 10 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 12 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 12 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 13 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 11 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 12 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 13 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 12 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 10 + 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 10 + 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 10 + 1
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 11 + 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 13 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 12 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 11 + 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 11 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12 + 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 12 + 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 11 + 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 12 + 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 10 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 12 + 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 11 + 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 12 + 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 13 + 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 12 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 10 + 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 10 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 10 + 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> ? = 11 + 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 12 + 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 12 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 11 + 1
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 12 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12 + 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 11 + 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 12 + 1
[1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 11 + 1
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 12 + 1
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 13 + 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 12 + 1
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000300
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000300: Graphs ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000300: Graphs ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Values
[1,0,1,0]
=> [1,2] => [2,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,0,0]
=> [2,1] => [1,2] => ([],2)
=> 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 5 = 4 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> 6 = 5 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => ([(1,2)],3)
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 7 = 6 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 8 = 7 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 7 = 6 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 8 = 7 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 9 = 8 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [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)
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9 = 8 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 9 = 8 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9 = 8 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7 = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9 = 8 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10 = 9 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 10 = 9 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 9 = 8 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 10 = 9 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 11 = 10 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 10 = 9 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => [6,5,7,3,4,1,2] => ([(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),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 11 + 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => [3,7,6,5,4,1,2] => ([(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)],7)
=> ? = 12 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => [6,5,3,7,4,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 12 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => [6,7,4,3,5,1,2] => ([(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),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 11 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6] => [6,4,7,3,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 12 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => [6,4,3,7,5,1,2] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 13 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,3,4,7,5] => [5,7,4,3,6,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 12 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,7,4,5] => [5,4,7,3,6,1,2] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 13 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,3,4,5] => [5,4,3,7,6,1,2] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 14 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => [6,5,4,3,7,1,2] => ([(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)],7)
=> ? = 12 + 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,6,7,4,5] => [5,4,7,6,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,4,5,6] => [6,5,4,7,1,3,2] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,5,6] => [6,5,7,1,4,3,2] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6] => [6,7,1,5,4,3,2] => ([(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)],7)
=> ? = 12 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,6,1,7,4,5] => [5,4,7,1,6,3,2] => ([(0,2),(0,3),(0,4),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 14 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,6,7,1,4,5] => [5,4,1,7,6,3,2] => ([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 15 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,1,3,7,5,6] => [6,5,7,3,1,4,2] => ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => [6,5,3,1,7,4,2] => ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 14 + 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,1,3,4,7,6] => [6,7,4,3,1,5,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 12 + 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,5,1,7,3,4,6] => [6,4,3,7,1,5,2] => ([(0,1),(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 14 + 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,1,3,4,6] => [6,4,3,1,7,5,2] => ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 15 + 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,6,1,3,7,4,5] => [5,4,7,3,1,6,2] => ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 14 + 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,6,1,7,3,4,5] => [5,4,3,7,1,6,2] => ([(0,1),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 15 + 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,1,3,4,5] => [5,4,3,1,7,6,2] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 16 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => [6,7,4,5,2,1,3] => ([(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),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 11 + 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [3,1,2,5,6,7,4] => [4,7,6,5,2,1,3] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,1,2,5,7,4,6] => [6,4,7,5,2,1,3] => ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,1,2,6,4,7,5] => [5,7,4,6,2,1,3] => ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,1,2,6,7,4,5] => [5,4,7,6,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13 + 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => [6,5,4,7,2,1,3] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,7,5,6] => [6,5,7,2,4,1,3] => ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [3,1,5,2,4,7,6] => [6,7,4,2,5,1,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 12 + 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,1,5,7,2,4,6] => [6,4,2,7,5,1,3] => ([(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 14 + 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,7,4,5] => [5,4,7,2,6,1,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 14 + 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,1,6,7,2,4,5] => [5,4,2,7,6,1,3] => ([(0,1),(0,3),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 15 + 1
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,4,1,2,6,7,5] => [5,7,6,2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13 + 1
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,1,2,7,5,6] => [6,5,7,2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13 + 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,4,1,7,2,5,6] => [6,5,2,7,1,4,3] => ([(0,2),(0,3),(0,4),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 14 + 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6] => [6,5,2,1,7,4,3] => ([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 15 + 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [3,5,1,2,4,7,6] => [6,7,4,2,1,5,3] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 13 + 1
[1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,1,2,6,7,4] => [4,7,6,2,1,5,3] => ([(0,2),(0,3),(0,4),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 14 + 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [3,5,1,2,7,4,6] => [6,4,7,2,1,5,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 14 + 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,1,7,2,4,6] => [6,4,2,7,1,5,3] => ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 15 + 1
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,1,2,4,7,5] => [5,7,4,2,1,6,3] => ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 14 + 1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,1,2,7,4,5] => [5,4,7,2,1,6,3] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ? = 15 + 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,1,7,2,4,5] => [5,4,2,7,1,6,3] => ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 16 + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,1,2,4,5] => [5,4,2,1,7,6,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 17 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => [6,5,4,2,1,7,3] => ([(0,3),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 14 + 1
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [4,1,2,3,6,7,5] => [5,7,6,3,2,1,4] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => [6,5,7,3,2,1,4] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12 + 1
Description
The number of independent sets of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is [[St000093]]
Matching statistic: St000395
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 78%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 78%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 9 = 8 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 9 = 8 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 10 = 9 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 10 = 9 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 9 = 8 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 10 = 9 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 11 = 10 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 10 = 9 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 9 + 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 10 + 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 11 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 10 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 12 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 11 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 9 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 11 + 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 10 + 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 11 + 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 12 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 12 + 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 10 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 11 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 12 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 12 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 13 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 11 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 12 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 13 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 14 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 12 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 9 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 10 + 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 10 + 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 11 + 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 10 + 1
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 11 + 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 13 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 12 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 10 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 11 + 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 11 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 12 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 11 + 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 12 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 12 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 13 + 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 13 + 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 12 + 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 13 + 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 11 + 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 12 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 14 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 15 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 13 + 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 10 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 12 + 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 13 + 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 14 + 1
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000070
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 89%
Mp00065: Permutations —permutation poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 89%
Values
[1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,0,0]
=> [2,1] => ([],2)
=> 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 5 = 4 + 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 6 = 5 + 1
[1,1,1,0,0,0]
=> [3,1,2] => ([(1,2)],3)
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 7 = 6 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 8 = 7 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 7 = 6 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 8 = 7 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 9 = 8 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 8 = 7 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 8 = 7 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 9 = 8 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9 = 8 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 8 = 7 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 9 = 8 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 7 = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 8 = 7 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 8 = 7 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 8 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 10 = 9 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 10 = 9 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 9 = 8 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 10 = 9 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 11 = 10 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 10 = 9 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 10 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 10 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 9 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 9 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
=> ? = 10 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> ? = 11 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => ([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
=> ? = 10 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => ([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => ([(0,5),(2,6),(3,2),(4,1),(4,6),(5,3),(5,4)],7)
=> ? = 12 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> ? = 11 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => ([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
=> ? = 12 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 9 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
=> ? = 10 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
=> ? = 10 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ? = 9 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
=> ? = 10 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> ? = 11 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,4,5,7] => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
=> ? = 10 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,4,5] => ([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
=> ? = 12 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> ? = 11 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => ([(0,4),(0,5),(1,6),(4,6),(5,1),(6,2),(6,3)],7)
=> ? = 10 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
=> ? = 10 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,5,6] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 11 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => ([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 11 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6] => ([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
=> ? = 12 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => ([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7)
=> ? = 12 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,2,7,4,5] => ([(0,3),(0,4),(2,5),(3,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ? = 13 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,4,5] => ([(0,3),(0,5),(3,6),(4,1),(5,4),(5,6),(6,2)],7)
=> ? = 14 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,2,4,5,6] => ([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7)
=> ? = 12 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,2,3,5,7,6] => ([(0,4),(0,5),(1,6),(4,6),(5,1),(6,2),(6,3)],7)
=> ? = 10 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,2,3,6,5,7] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
=> ? = 10 + 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,2,3,6,7,5] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 11 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,2,3,7,5,6] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 11 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => ([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7)
=> ? = 12 + 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,2,6,3,5,7] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> ? = 11 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,2,6,7,3,5] => ([(0,3),(0,4),(2,5),(3,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ? = 13 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,7,3,5,6] => ([(0,3),(0,4),(2,6),(3,5),(3,6),(4,2),(4,5),(6,1)],7)
=> ? = 12 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,2,3,7,6] => ([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1)],7)
=> ? = 12 + 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,2,7,3,6] => ([(0,3),(0,4),(1,5),(2,5),(2,6),(3,2),(4,1),(4,6)],7)
=> ? = 13 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,7,2,3,6] => ([(0,4),(0,5),(2,6),(3,1),(3,6),(4,2),(5,3)],7)
=> ? = 14 + 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => ([(0,3),(0,4),(1,5),(2,5),(2,6),(3,2),(4,1),(4,6)],7)
=> ? = 13 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,2,3,5,6] => ([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
=> ? = 13 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,2,3,4,7,6] => ([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 11 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7)
=> ? = 12 + 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,2,3,7,4,6] => ([(0,2),(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,3)],7)
=> ? = 12 + 1
Description
The number of antichains in a poset.
An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable.
An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St000394
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 89%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 89%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 6 = 5 + 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 9 = 8 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 10 = 9 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 10 = 9 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> 10 = 9 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> 11 = 10 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> 10 = 9 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0,0]
=> ? = 12 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,0]
=> ? = 12 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0]
=> ? = 13 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 9 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 10 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 10 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 10 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 11 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 10 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 12 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 11 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 10 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 10 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 10 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 11 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 11 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 12 + 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001213
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001213: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 78%
St001213: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 78%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 7 = 6 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 8 = 7 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 9 = 8 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 10 = 9 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 10 = 9 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 10 = 9 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 11 = 10 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 10 = 9 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 11 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 9 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 11 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 10 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 12 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 11 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 12 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 13 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 11 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 9 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 10 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 10 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 9 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 10 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 11 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 10 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 12 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 11 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 9 + 1
Description
The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.
Matching statistic: St000180
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000180: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 78%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000180: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 78%
Values
[1,0,1,0]
=> [1,2] => [2,1] => ([],2)
=> 3 = 2 + 1
[1,1,0,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 5 = 4 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 6 = 5 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 7 = 6 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 8 = 7 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 7 = 6 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 8 = 7 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 9 = 8 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 8 = 7 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 8 = 7 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 9 = 8 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 9 = 8 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 9 = 8 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 7 = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 8 = 7 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 8 = 7 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> 9 = 8 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 10 = 9 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 10 = 9 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 9 = 8 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 10 = 9 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 11 = 10 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 10 = 9 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 8 = 7 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [4,6,5,2,3,1] => ([(1,5),(2,3),(2,4)],6)
=> ? = 9 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => [5,6,2,4,3,1] => ([(1,5),(2,3),(2,4)],6)
=> ? = 9 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => [5,2,6,4,3,1] => ([(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 10 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => [4,6,2,5,3,1] => ([(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 10 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => [4,2,6,5,3,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 11 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => [5,4,2,6,3,1] => ([(1,5),(2,5),(3,4),(3,5)],6)
=> ? = 10 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => [3,6,5,2,4,1] => ([(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 10 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => [3,6,2,5,4,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 11 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => [4,6,3,2,5,1] => ([(1,5),(2,5),(3,4),(3,5)],6)
=> ? = 10 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [4,6,5,3,1,2] => ([(1,5),(2,3),(2,4)],6)
=> ? = 9 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [6,3,5,4,1,2] => ([(1,5),(2,3),(2,4)],6)
=> ? = 9 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => [3,6,5,4,1,2] => ([(0,5),(1,2),(1,3),(1,4)],6)
=> ? = 10 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> ? = 11 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => [5,6,4,1,3,2] => ([(1,5),(2,3),(2,4)],6)
=> ? = 9 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => [6,4,5,1,3,2] => ([(1,5),(2,3),(2,4)],6)
=> ? = 9 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,4,5] => [5,4,6,1,3,2] => ([(0,5),(1,5),(2,3),(2,4)],6)
=> ? = 10 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => [5,6,1,4,3,2] => ([(0,5),(1,2),(1,3),(1,4)],6)
=> ? = 10 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 11 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => [6,4,1,5,3,2] => ([(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 10 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [4,6,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
=> ? = 11 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 12 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 11 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => [6,3,5,1,4,2] => ([(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 10 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
=> ? = 11 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,1,3,6] => [6,3,1,5,4,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 11 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 13 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [6,4,3,1,5,2] => ([(1,5),(2,5),(3,4),(3,5)],6)
=> ? = 10 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [4,6,3,1,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 11 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 12 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 13 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [5,4,3,1,6,2] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ? = 11 + 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => [4,6,5,2,1,3] => ([(0,5),(1,5),(2,3),(2,4)],6)
=> ? = 10 + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => [6,2,5,4,1,3] => ([(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 10 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 11 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6)
=> ? = 11 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 12 + 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [3,4,1,2,6,5] => [5,6,2,1,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> ? = 11 + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,1,5,2,6] => [6,2,5,1,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 11 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 12 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => [5,2,6,1,4,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 12 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 13 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 13 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 12 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 13 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => [4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 14 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5] => [5,4,2,1,6,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 12 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => [6,3,5,2,1,4] => ([(1,5),(2,5),(3,4),(3,5)],6)
=> ? = 10 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 11 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 13 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 13 + 1
Description
The number of chains of a poset.
Matching statistic: St000081
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 56%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 56%
Values
[1,0,1,0]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 6
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 8
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 9
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 10
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 12
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 12
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 11
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
Description
The number of edges of a graph.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001616The number of neutral elements in a lattice. St000019The cardinality of the support of a permutation. St001428The number of B-inversions of a signed permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000029The depth of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
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