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Your data matches 20 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000180: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 2
[1,0,1,0]
=> [1,2] => [2,1] => ([],2)
=> 3
[1,1,0,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 4
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 4
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 5
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 5
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 6
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 6
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 5
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 6
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 6
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 7
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 7
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 6
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 7
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 7
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 8
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 8
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 7
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 8
[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
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 8
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 7
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 7
[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
[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
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 7
[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
[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
[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
[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
[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
[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
[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
[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
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 7
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
Description
The number of chains of a poset.
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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000300: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 2
[1,0,1,0]
=> [1,2] => [2,1] => ([(0,1)],2)
=> 3
[1,1,0,0]
=> [2,1] => [1,2] => ([],2)
=> 4
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 5
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 5
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> 6
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => ([(1,2)],3)
=> 6
[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
[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
[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
[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
[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
[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
[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
[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
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 8
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 8
[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
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 8
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 9
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 8
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,1,4,6,7,5,8] => [8,5,7,6,4,1,3,2] => ?
=> ? = 13
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4,7,8] => [8,7,4,6,5,1,3,2] => ?
=> ? = 13
[1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [5,6,1,2,3,4,7,8] => [8,7,4,3,2,1,6,5] => ?
=> ? = 17
Description
The number of independent sets of vertices of a graph.
An independent set of vertices of a graph G is a subset U⊂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↦V(G)∖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: 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: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 95%
Mp00065: Permutations —permutation poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 95%
Values
[1,0]
=> [1] => ([],1)
=> 2
[1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> 3
[1,1,0,0]
=> [2,1] => ([],2)
=> 4
[1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 4
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 5
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 5
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 6
[1,1,1,0,0,0]
=> [3,1,2] => ([(1,2)],3)
=> 6
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 6
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 7
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 7
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 6
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 7
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 7
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 8
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 8
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 7
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 8
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 9
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 8
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 10
[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
[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
[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
[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
[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
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => ([(0,4),(1,6),(1,7),(2,6),(2,7),(3,5),(4,3),(5,1),(5,2)],8)
=> ? = 11
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => ([(0,6),(3,7),(4,2),(5,1),(6,3),(7,4),(7,5)],8)
=> ? = 13
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => ([(0,5),(1,7),(2,7),(5,6),(6,1),(6,2),(7,3),(7,4)],8)
=> ? = 11
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2),(5,7),(6,7)],8)
=> ? = 11
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,4,5,8] => ([(0,5),(1,7),(2,7),(3,2),(4,1),(5,6),(6,3),(6,4)],8)
=> ? = 13
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,8,7] => ([(0,6),(3,7),(4,7),(5,1),(5,2),(6,3),(6,4),(7,5)],8)
=> ? = 11
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(5,7),(6,7),(7,1)],8)
=> ? = 11
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,4,5,3,7,8,6] => ([(0,5),(2,6),(2,7),(3,6),(3,7),(4,3),(5,2),(5,4),(7,1)],8)
=> ? = 13
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ? = 15
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,3,4,8,6,7] => ([(0,5),(2,6),(2,7),(3,6),(3,7),(4,3),(5,2),(5,4),(7,1)],8)
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,2,5,6,3,4,7,8] => ([(0,6),(1,7),(2,7),(4,2),(5,1),(6,4),(6,5),(7,3)],8)
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,8,3,4] => ([(0,7),(3,5),(4,3),(5,2),(6,1),(7,4),(7,6)],8)
=> ? = 17
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,6,3,8,4,5,7] => ([(0,5),(1,6),(2,6),(2,7),(3,1),(4,3),(4,7),(5,2),(5,4)],8)
=> ? = 15
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,7,8,3,4,5,6] => ([(0,7),(3,5),(4,3),(5,2),(6,1),(7,4),(7,6)],8)
=> ? = 17
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,6,8,7] => ([(0,3),(0,4),(3,7),(4,7),(5,6),(6,1),(6,2),(7,5)],8)
=> ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(5,7),(6,7),(7,4)],8)
=> ? = 11
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => ([(0,3),(0,4),(1,6),(2,5),(2,7),(3,5),(3,7),(4,2),(5,6),(7,1)],8)
=> ? = 13
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,3,5,6,2,7,4,8] => ([(0,2),(0,4),(1,5),(2,5),(2,6),(3,1),(4,3),(4,6),(5,7),(6,7)],8)
=> ? = 15
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,4,2,3,5,8,6,7] => ([(0,4),(0,6),(2,7),(4,7),(5,1),(6,2),(7,3),(7,5)],8)
=> ? = 13
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,7,5,6,8] => ([(0,3),(0,4),(1,6),(2,5),(2,7),(3,5),(3,7),(4,2),(5,6),(7,1)],8)
=> ? = 13
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,4,5,2,3,6,7,8] => ([(0,5),(0,6),(2,7),(3,7),(4,1),(5,3),(6,2),(7,4)],8)
=> ? = 13
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,7,2,3,8] => ([(0,5),(0,6),(1,7),(2,7),(3,4),(4,2),(5,3),(6,1)],8)
=> ? = 17
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,7,3,4,6,8] => ([(0,2),(0,4),(1,5),(2,5),(2,6),(3,1),(4,3),(4,6),(5,7),(6,7)],8)
=> ? = 15
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,6,7,2,3,4,5,8] => ([(0,5),(0,6),(1,7),(2,7),(3,4),(4,2),(5,3),(6,1)],8)
=> ? = 17
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,6,8,7] => ([(0,7),(1,7),(4,6),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 11
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,5,7,6,8] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,2),(5,3),(7,4)],8)
=> ? = 11
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,4,6,5,7,8] => ([(0,6),(1,6),(3,7),(4,7),(5,3),(5,4),(6,5),(7,2)],8)
=> ? = 11
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ([(0,7),(1,7),(3,6),(4,6),(5,2),(6,5),(7,3),(7,4)],8)
=> ? = 11
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7,8] => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,7),(6,7),(7,3)],8)
=> ? = 11
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,5,7,8,6] => ([(0,7),(1,4),(4,7),(5,3),(6,2),(6,5),(7,6)],8)
=> ? = 13
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,1,4,6,7,5,8] => ([(0,7),(1,2),(2,7),(3,6),(4,6),(5,4),(7,3),(7,5)],8)
=> ? = 13
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4,7,8] => ([(0,5),(0,7),(1,2),(2,5),(2,7),(3,6),(5,6),(6,4),(7,3)],8)
=> ? = 13
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,4,1,6,7,8,5] => ([(0,6),(0,7),(1,4),(3,6),(3,7),(4,3),(5,2),(7,5)],8)
=> ? = 15
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,5,6,1,7,8,4] => ([(0,6),(0,7),(1,5),(3,7),(4,3),(5,4),(5,6),(7,2)],8)
=> ? = 17
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,6,8,5,7] => ([(0,7),(1,3),(1,7),(3,5),(3,6),(5,4),(6,2),(6,4),(7,5),(7,6)],8)
=> ? = 15
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,3,7,5,8,6] => ([(0,7),(1,3),(1,7),(3,5),(3,6),(5,4),(6,2),(6,4),(7,5),(7,6)],8)
=> ? = 15
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [2,4,5,1,6,3,7,8] => ([(0,5),(0,6),(1,4),(1,6),(3,5),(4,3),(5,7),(6,7),(7,2)],8)
=> ? = 15
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,1,8,3] => ([(0,6),(0,7),(1,5),(1,7),(2,6),(3,4),(4,2),(5,3)],8)
=> ? = 19
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [2,5,1,7,3,4,8,6] => ([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ? = 17
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,6,7,1,3,4,8,5] => ([(0,7),(1,4),(1,7),(3,6),(4,3),(5,2),(5,6),(7,5)],8)
=> ? = 19
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,5,8,6,7] => ([(0,7),(1,4),(4,7),(5,3),(6,2),(6,5),(7,6)],8)
=> ? = 13
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,4,2,6,8,5,7] => ([(0,7),(1,3),(1,7),(3,5),(3,6),(5,4),(6,2),(6,4),(7,5),(7,6)],8)
=> ? = 15
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,1,4,2,7,5,8,6] => ([(0,7),(1,3),(1,7),(3,5),(3,6),(5,4),(6,2),(6,4),(7,5),(7,6)],8)
=> ? = 15
[1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,5,6,2,8,4,7] => ([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ? = 17
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,2,5,6,7,8] => ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ? = 13
[1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,4,5,6,1,2,7,8] => ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ? = 17
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,8,1,2] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? = 21
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [3,6,7,1,2,8,4,5] => ([(0,4),(1,5),(1,7),(3,6),(4,6),(4,7),(5,3),(7,2)],8)
=> ? = 21
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,1,2,3,8,5,6,7] => ([(0,6),(0,7),(1,4),(3,6),(3,7),(4,3),(5,2),(7,5)],8)
=> ? = 15
[1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [4,1,5,6,8,2,3,7] => ([(0,7),(1,4),(1,7),(3,6),(4,3),(5,2),(5,6),(7,5)],8)
=> ? = 19
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∈I and b≤Pa implies b∈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)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 85%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 85%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[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
[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
[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
[1,1,0,1,0,0]
=> [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
[1,1,1,0,0,0]
=> [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
[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
[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
[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
[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]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 7
[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]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 7
[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
[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
[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]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7
[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]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 8
[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]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 8
[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]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 7
[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]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 8
[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
[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]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 8
[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
[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
[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
[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]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 8
[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]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 8
[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
[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
[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]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 8
[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]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 9
[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]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 9
[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]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> 8
[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]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> 9
[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
[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]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 9
[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
[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
[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
[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]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> 9
[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]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 9
[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]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 8
[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]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> 9
[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]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 9
[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]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 10
[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]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> 10
[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]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> 9
[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]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> 10
[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]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> 11
[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]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> 10
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,0,0]
=> ? = 11
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0,0,0]
=> ? = 13
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,0]
=> ?
=> ? = 11
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,0]
=> ?
=> ? = 11
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [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]
=> ?
=> ? = 13
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 11
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 11
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 11
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [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]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 13
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [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]
=> ?
=> ? = 15
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ?
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [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]
=> [1,1,1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [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]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0,0]
=> ? = 17
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ?
=> ? = 15
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 17
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 11
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ?
=> ? = 11
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ?
=> ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ?
=> ? = 11
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [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]
=> ?
=> ? = 13
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 15
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ?
=> ? = 13
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ?
=> ? = 13
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [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]
=> ?
=> ? = 13
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 17
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ?
=> ? = 15
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [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]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 17
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? = 11
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0]
=> ? = 11
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 11
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ?
=> ? = 11
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 11
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 13
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> ?
=> ? = 15
[1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 15
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> ? = 13
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0]
=> ? = 13
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 13
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> ? = 15
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? = 17
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 15
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ?
=> ? = 15
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 15
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 19
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 17
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 19
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ?
=> ? = 13
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> ?
=> ? = 15
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> ?
=> ? = 15
[1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 17
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001034
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,1,0,0]
=> 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 6
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,0]
=> [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]
=> 5
[1,0,1,0,1,1,0,0]
=> [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]
=> 6
[1,0,1,1,0,0,1,0]
=> [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]
=> 6
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[1,1,0,0,1,0,1,0]
=> [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]
=> 6
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 7
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 7
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 8
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 8
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 9
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 8
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 7
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [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]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [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]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 8
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 9
[1,1,0,0,1,0,1,0,1,0]
=> [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]
=> 7
[1,1,0,0,1,0,1,1,0,0]
=> [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]
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [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]
=> 8
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 9
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 8
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 10
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 11
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 10
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 11
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 11
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 11
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 13
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 11
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 11
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 11
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 13
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 15
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 17
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 15
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 17
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 11
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 11
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 11
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 13
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 15
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 13
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 13
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 13
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 17
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 15
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 17
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 11
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 11
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 11
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 11
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 11
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 13
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 15
[1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 15
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 13
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 13
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 13
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 15
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 17
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 15
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 15
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 15
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? = 19
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 17
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 19
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [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]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 13
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 15
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 15
[1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [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]
=> ? = 17
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 13
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
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: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 70%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [1] => ([],1)
=> ? = 2 - 1
[1,0,1,0]
=> [1,2] => [2,1] => ([],2)
=> 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 3 = 4 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 4 = 5 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 4 = 5 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 5 = 6 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 6 = 7 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 6 = 7 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 5 = 6 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6 = 7 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 6 = 7 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 7 = 8 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 7 = 8 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 6 = 7 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 7 = 8 - 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)
=> 8 = 9 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7 = 8 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 6 = 7 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 6 = 7 - 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)
=> 7 = 8 - 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)
=> 7 = 8 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 6 = 7 - 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)
=> 7 = 8 - 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)
=> 7 = 8 - 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)
=> 8 = 9 - 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)
=> 8 = 9 - 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)
=> 7 = 8 - 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)
=> 8 = 9 - 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)
=> 9 = 10 - 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)
=> 8 = 9 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 6 = 7 - 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)
=> 7 = 8 - 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)
=> 7 = 8 - 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)
=> 8 = 9 - 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)
=> 8 = 9 - 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)
=> 7 = 8 - 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)
=> 8 = 9 - 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)
=> 8 = 9 - 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)
=> 9 = 10 - 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)
=> 9 = 10 - 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)
=> 8 = 9 - 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)
=> 9 = 10 - 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)
=> 10 = 11 - 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)
=> 9 = 10 - 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)
=> 7 = 8 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => [7,8,5,6,4,3,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => [6,5,8,7,4,3,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 13 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => [7,8,6,4,5,3,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [8,6,7,4,5,3,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,4,5,8] => [8,5,4,7,6,3,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 13 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,8,7] => [7,8,6,5,3,4,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5,7,6,8] => [8,6,7,5,3,4,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [8,7,5,6,3,4,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,4,5,3,7,8,6] => [6,8,7,3,5,4,2,1] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => [5,8,3,7,6,4,2,1] => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 15 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,3,4,8,6,7] => [7,6,8,4,3,5,2,1] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 13 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,2,5,6,3,4,7,8] => [8,7,4,3,6,5,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 13 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,8,3,4] => [4,3,8,7,6,5,2,1] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 17 - 1
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,6,3,8,4,5,7] => [7,5,4,8,3,6,2,1] => ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 15 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,7,8,3,4,5,6] => [6,5,4,3,8,7,2,1] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 17 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,6,8,7] => [7,8,6,5,4,2,3,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => [8,6,7,5,4,2,3,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,6,5,7,8] => [8,7,5,6,4,2,3,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => [8,7,6,4,5,2,3,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [8,5,7,6,2,4,3,1] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,3,5,6,2,7,4,8] => [8,4,7,2,6,5,3,1] => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 15 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,4,2,3,5,8,6,7] => [7,6,8,5,3,2,4,1] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 13 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,7,5,6,8] => [8,6,5,7,3,2,4,1] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 13 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,4,5,2,3,6,7,8] => [8,7,6,3,2,5,4,1] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 13 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,7,2,3,8] => [8,3,2,7,6,5,4,1] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 17 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,7,3,4,6,8] => [8,6,4,3,7,2,5,1] => ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 15 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,6,7,2,3,4,5,8] => [8,5,4,3,2,7,6,1] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 17 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,6,8,7] => [7,8,6,5,4,3,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,5,7,6,8] => [8,6,7,5,4,3,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,4,6,5,7,8] => [8,7,5,6,4,3,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => [8,7,6,4,5,3,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7,8] => [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 13 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,4,3,7,8,5,6] => [6,5,8,7,3,4,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ? = 15 - 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,1,5,6,3,4,8,7] => [7,8,4,3,6,5,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ? = 15 - 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,5,7,8,6] => [6,8,7,5,4,1,3,2] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,1,4,6,7,5,8] => [8,5,7,6,4,1,3,2] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4,7,8] => [8,7,4,6,5,1,3,2] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,4,1,6,7,8,5] => [5,8,7,6,1,4,3,2] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4)],8)
=> ? = 15 - 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,5,6,1,7,8,4] => [4,8,7,1,6,5,3,2] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7)],8)
=> ? = 17 - 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,6,8,5,7] => [7,5,8,6,3,1,4,2] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8)
=> ? = 15 - 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,3,7,5,8,6] => [6,8,5,7,3,1,4,2] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8)
=> ? = 15 - 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [2,4,5,1,6,3,7,8] => [8,7,3,6,1,5,4,2] => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 15 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,1,8,3] => [3,8,1,7,6,5,4,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7)],8)
=> ? = 19 - 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [2,5,1,7,3,4,8,6] => [6,8,4,3,7,1,5,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6)],8)
=> ? = 17 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,6,7,1,3,4,8,5] => [5,8,4,3,1,7,6,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 19 - 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,5,8,6,7] => [7,6,8,5,4,2,1,3] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 13 - 1
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,4,2,6,8,5,7] => [7,5,8,6,2,4,1,3] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8)
=> ? = 15 - 1
Description
The number of relations in a poset.
This is the number of intervals x,y with x≤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: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 70%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000641: Posets ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [1] => ([],1)
=> ? = 2 - 1
[1,0,1,0]
=> [1,2] => [2,1] => ([],2)
=> 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 3 = 4 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 4 = 5 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 4 = 5 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 5 = 6 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 6 = 7 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 6 = 7 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 5 = 6 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6 = 7 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 6 = 7 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 7 = 8 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 7 = 8 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 6 = 7 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 7 = 8 - 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)
=> 8 = 9 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7 = 8 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 6 = 7 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 6 = 7 - 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)
=> 7 = 8 - 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)
=> 7 = 8 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 6 = 7 - 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)
=> 7 = 8 - 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)
=> 7 = 8 - 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)
=> 8 = 9 - 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)
=> 8 = 9 - 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)
=> 7 = 8 - 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)
=> 8 = 9 - 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)
=> 9 = 10 - 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)
=> 8 = 9 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 6 = 7 - 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)
=> 7 = 8 - 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)
=> 7 = 8 - 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)
=> 8 = 9 - 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)
=> 8 = 9 - 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)
=> 7 = 8 - 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)
=> 8 = 9 - 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)
=> 8 = 9 - 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)
=> 9 = 10 - 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)
=> 9 = 10 - 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)
=> 8 = 9 - 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)
=> 9 = 10 - 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)
=> 10 = 11 - 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)
=> 9 = 10 - 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)
=> 7 = 8 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => [7,8,5,6,4,3,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => [6,5,8,7,4,3,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 13 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => [7,8,6,4,5,3,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [8,6,7,4,5,3,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,4,5,8] => [8,5,4,7,6,3,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 13 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,8,7] => [7,8,6,5,3,4,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5,7,6,8] => [8,6,7,5,3,4,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [8,7,5,6,3,4,2,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,4,5,3,7,8,6] => [6,8,7,3,5,4,2,1] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => [5,8,3,7,6,4,2,1] => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 15 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,3,4,8,6,7] => [7,6,8,4,3,5,2,1] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 13 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,2,5,6,3,4,7,8] => [8,7,4,3,6,5,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 13 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,8,3,4] => [4,3,8,7,6,5,2,1] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 17 - 1
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,6,3,8,4,5,7] => [7,5,4,8,3,6,2,1] => ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 15 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,7,8,3,4,5,6] => [6,5,4,3,8,7,2,1] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 17 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,6,8,7] => [7,8,6,5,4,2,3,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => [8,6,7,5,4,2,3,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,6,5,7,8] => [8,7,5,6,4,2,3,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => [8,7,6,4,5,2,3,1] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [8,5,7,6,2,4,3,1] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,3,5,6,2,7,4,8] => [8,4,7,2,6,5,3,1] => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 15 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,4,2,3,5,8,6,7] => [7,6,8,5,3,2,4,1] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 13 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,7,5,6,8] => [8,6,5,7,3,2,4,1] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 13 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,4,5,2,3,6,7,8] => [8,7,6,3,2,5,4,1] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 13 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,7,2,3,8] => [8,3,2,7,6,5,4,1] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 17 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,7,3,4,6,8] => [8,6,4,3,7,2,5,1] => ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 15 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,6,7,2,3,4,5,8] => [8,5,4,3,2,7,6,1] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 17 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,6,8,7] => [7,8,6,5,4,3,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,5,7,6,8] => [8,6,7,5,4,3,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,4,6,5,7,8] => [8,7,5,6,4,3,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => [8,7,6,4,5,3,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7,8] => [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
=> ? = 11 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 13 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,4,3,7,8,5,6] => [6,5,8,7,3,4,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ? = 15 - 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,1,5,6,3,4,8,7] => [7,8,4,3,6,5,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ? = 15 - 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,5,7,8,6] => [6,8,7,5,4,1,3,2] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,1,4,6,7,5,8] => [8,5,7,6,4,1,3,2] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4,7,8] => [8,7,4,6,5,1,3,2] => ([(2,6),(2,7),(3,4),(3,5)],8)
=> ? = 13 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,4,1,6,7,8,5] => [5,8,7,6,1,4,3,2] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4)],8)
=> ? = 15 - 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,5,6,1,7,8,4] => [4,8,7,1,6,5,3,2] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7)],8)
=> ? = 17 - 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,6,8,5,7] => [7,5,8,6,3,1,4,2] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8)
=> ? = 15 - 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,3,7,5,8,6] => [6,8,5,7,3,1,4,2] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8)
=> ? = 15 - 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [2,4,5,1,6,3,7,8] => [8,7,3,6,1,5,4,2] => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 15 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,1,8,3] => [3,8,1,7,6,5,4,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7)],8)
=> ? = 19 - 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [2,5,1,7,3,4,8,6] => [6,8,4,3,7,1,5,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6)],8)
=> ? = 17 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,6,7,1,3,4,8,5] => [5,8,4,3,1,7,6,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ? = 19 - 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,5,8,6,7] => [7,6,8,5,4,2,1,3] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 13 - 1
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,4,2,6,8,5,7] => [7,5,8,6,2,4,1,3] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8)
=> ? = 15 - 1
Description
The number of non-empty boolean intervals in a poset.
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: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
St001213: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> 2
[1,0,1,0]
=> [1,1,0,0]
=> 3
[1,1,0,0]
=> [1,0,1,0]
=> 4
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 5
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 5
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 6
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 6
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 6
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 7
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 6
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 7
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 7
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 8
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 8
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 7
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 8
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 9
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 8
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 7
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 8
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 9
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 7
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 8
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 9
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 8
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 11
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 14
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 14
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 11
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 13
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 11
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> ? = 13
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 11
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 11
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 11
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 13
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 15
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 17
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> ? = 15
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 17
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 11
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 11
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 11
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 13
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 15
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 13
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 13
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 13
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 17
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 15
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> ? = 17
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 11
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 11
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 11
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 11
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 11
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 13
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 15
[1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 15
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 13
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 13
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 13
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 15
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 17
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 15
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 15
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 15
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 19
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 17
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 19
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 13
Description
The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.
Matching statistic: St000395
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 4
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 6
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 6
[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]
=> 5
[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]
=> 6
[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]
=> 6
[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]
=> 7
[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]
=> 7
[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]
=> 6
[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]
=> 7
[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]
=> 7
[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]
=> 8
[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]
=> 8
[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]
=> 7
[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]
=> 8
[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]
=> 9
[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]
=> 8
[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]
=> 6
[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]
=> 7
[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]
=> 7
[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]
=> 8
[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]
=> 8
[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]
=> 7
[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]
=> 8
[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]
=> 8
[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]
=> 9
[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]
=> 9
[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]
=> 8
[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]
=> 9
[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]
=> 10
[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]
=> 9
[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]
=> 7
[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]
=> 8
[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]
=> 8
[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]
=> 9
[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]
=> 9
[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]
=> 8
[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]
=> 9
[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]
=> 9
[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]
=> 10
[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]
=> 10
[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]
=> 9
[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]
=> 10
[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]
=> 11
[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]
=> 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 14
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 14
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 11
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [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]
=> ? = 13
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 11
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 11
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [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]
=> ? = 13
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 11
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 11
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 11
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 13
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 15
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [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]
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [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]
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 17
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [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]
=> ? = 15
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [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]
=> ? = 17
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 11
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 11
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 11
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 13
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 15
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 13
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [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]
=> ? = 13
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [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]
=> ? = 13
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [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]
=> ? = 17
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 15
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 17
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 11
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 11
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 11
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 11
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 11
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 13
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 15
[1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 15
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 13
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 13
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 13
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 15
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 17
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 15
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 15
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 15
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 19
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 17
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 19
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 13
Description
The sum of the heights of the peaks of a Dyck path.
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: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 55%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 55%
Values
[1,0]
=> [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[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 = 6 - 1
[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 = 6 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[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 = 6 - 1
[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 = 6 - 1
[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 = 7 - 1
[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 = 7 - 1
[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 = 6 - 1
[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 = 7 - 1
[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 = 7 - 1
[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 = 8 - 1
[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 = 8 - 1
[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 = 7 - 1
[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 = 8 - 1
[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 = 9 - 1
[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 = 8 - 1
[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 = 6 - 1
[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 = 7 - 1
[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 = 7 - 1
[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 = 8 - 1
[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 = 8 - 1
[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 = 7 - 1
[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 = 8 - 1
[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 = 8 - 1
[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 = 9 - 1
[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 = 9 - 1
[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 = 8 - 1
[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 = 9 - 1
[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 = 10 - 1
[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 = 9 - 1
[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 = 7 - 1
[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 = 8 - 1
[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 = 8 - 1
[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 = 9 - 1
[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 = 9 - 1
[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 = 8 - 1
[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 = 9 - 1
[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 = 9 - 1
[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 = 10 - 1
[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 = 10 - 1
[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 = 9 - 1
[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 = 10 - 1
[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 = 11 - 1
[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 = 10 - 1
[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)
=> ? = 9 - 1
[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)
=> ? = 10 - 1
[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)
=> ? = 10 - 1
[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)
=> ? = 11 - 1
[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)
=> ? = 9 - 1
[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)
=> ? = 9 - 1
[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)
=> ? = 10 - 1
[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)
=> ? = 10 - 1
[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)
=> ? = 10 - 1
[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)
=> ? = 10 - 1
[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)
=> ? = 11 - 1
[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)
=> ? = 11 - 1
[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)
=> ? = 13 - 1
[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)
=> ? = 12 - 1
[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)
=> ? = 12 - 1
[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)
=> ? = 13 - 1
[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)
=> ? = 9 - 1
[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)
=> ? = 9 - 1
[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)
=> ? = 10 - 1
[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)
=> ? = 9 - 1
[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)
=> ? = 11 - 1
[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)
=> ? = 10 - 1
[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)
=> ? = 11 - 1
[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)
=> ? = 11 - 1
[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)
=> ? = 11 - 1
[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)
=> ? = 10 - 1
[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)
=> ? = 11 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 12 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 13 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => ([(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)
=> ? = 10 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,2,5,4,6] => ([(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)
=> ? = 10 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(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)
=> ? = 11 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => ([(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)
=> ? = 10 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => ([(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)
=> ? = 11 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,6] => ([(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)
=> ? = 11 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 12 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => ([(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)
=> ? = 12 - 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,1,2,5,6] => ([(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)
=> ? = 11 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => ([(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)
=> ? = 13 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(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)
=> ? = 15 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => ([(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)
=> ? = 12 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => ([(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)
=> ? = 11 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => ([(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)
=> ? = 12 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => ([(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)
=> ? = 12 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 13 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => ([(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)
=> ? = 13 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 16 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(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)
=> ? = 14 - 1
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. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000019The cardinality of the support of a permutation. St001428The number of B-inversions of a signed permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the 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 τΩ1 composed with its inverse in the corresponding Nakayama algebra.
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