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Your data matches 633 different statistics following compositions of up to 3 maps.
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Matching statistic: St000205
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],.]
=> [1,2] => [1,1]
=> [1]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1]
=> [1]
=> 0
[[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> [1]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> [1]
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [2,2]
=> [2]
=> 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1]
=> [1]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> [1]
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> [1]
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> [2]
=> 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2]
=> [2]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,1,1]
=> [1,1]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [4,1]
=> [1]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2]
=> [2]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [4,1]
=> [1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [4,1]
=> [1]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [4,1]
=> [1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [2,2,1]
=> [2,1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,1]
=> [1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St001577
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001577: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001577: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Values
[[.,.],.]
=> [1,2] => ([],2)
=> ([],1)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[[.,.],.],.]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [2,4,7,6,5,3,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [2,6,5,4,7,3,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [3,2,7,6,5,4,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [4,3,2,7,6,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [2,4,3,7,6,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [3,2,4,7,6,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [5,4,3,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [3,5,4,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [4,3,5,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [2,5,4,3,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [3,2,5,4,7,6,1] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [4,3,2,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[[.,.],[.,[.,.]]]],.]]
=> [3,6,5,4,2,7,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [3,2,6,5,4,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [4,3,2,6,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,5,7,6,4,3,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,6,5,7,4,3,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,4,7,6,5,3,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,6,5,4,7,3,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,3,7,6,5,4,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,3,5,7,6,4,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,3,6,5,7,4,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,3,5,4,7,6,2] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,4,3,5,7,6,2] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,6,5,4,3,7,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [1,4,6,5,3,7,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
Description
The minimal number of edges to add or remove to make a graph a cograph.
A cograph is a graph that can be obtained from the one vertex graph by complementation and disjoint union.
Matching statistic: St000264
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 24%●distinct values known / distinct values provided: 17%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 24%●distinct values known / distinct values provided: 17%
Values
[[.,.],.]
=> [1,2] => [1,2] => ([],2)
=> ? = 0 + 3
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 0 + 3
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 0 + 3
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0 + 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 0 + 3
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 0 + 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 0 + 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 + 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 0 + 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 3
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 0 + 3
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 0 + 3
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 0 + 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 3
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 0 + 3
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 0 + 3
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 3
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 0 + 3
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 0 + 3
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 + 3
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [6,1,2,4,5,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => [6,1,3,2,5,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[.,[[[.,.],.],[.,.]]]]
=> [3,4,6,5,2,1] => [6,1,3,4,2,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [6,1,5,3,4,2] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [6,1,5,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[.,[[[.,.],[.,.]],.]]]
=> [3,5,4,6,2,1] => [6,1,3,5,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [6,1,4,3,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,.],[.,[.,[.,.]]]]]
=> [2,6,5,4,3,1] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,.],[.,[[.,.],.]]]]
=> [2,5,6,4,3,1] => [6,2,1,3,5,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,.],[[[.,.],.],.]]]
=> [2,4,5,6,3,1] => [6,2,1,4,5,3] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,[.,.]],[.,[.,.]]]]
=> [3,2,6,5,4,1] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,[.,.]],[[.,.],.]]]
=> [3,2,5,6,4,1] => [6,3,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[[.,.],.],[.,[.,.]]]]
=> [2,3,6,5,4,1] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,[.,[.,.]]],[.,.]]]
=> [4,3,2,6,5,1] => [6,4,2,3,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,[[.,.],.]],[.,.]]]
=> [3,4,2,6,5,1] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[[.,.],[.,.]],[.,.]]]
=> [2,4,3,6,5,1] => [6,2,4,3,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[[.,[.,.]],.],[.,.]]]
=> [3,2,4,6,5,1] => [6,3,2,4,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[[[.,.],.],.],[.,.]]]
=> [2,3,4,6,5,1] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,[.,[.,[.,.]]]],.]]
=> [5,4,3,2,6,1] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,[.,[[.,.],.]]],.]]
=> [4,5,3,2,6,1] => [6,5,2,4,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[.,[[[.,.],.],.]],.]]
=> [3,4,5,2,6,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[[.,.],[.,[.,.]]],.]]
=> [2,5,4,3,6,1] => [6,2,5,3,4,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[.,[[[.,[.,.]],[.,.]],.]]
=> [3,2,5,4,6,1] => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,6,5,4,3,2] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[[.,[.,.]],[.,[.,[.,.]]]]
=> [2,1,6,5,4,3] => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[[[.,.],.],[.,[.,[.,.]]]]
=> [1,2,6,5,4,3] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[[.,[.,[.,.]]],[.,[.,.]]]
=> [3,2,1,6,5,4] => [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? = 0 + 3
[[[.,.],[.,.]],[.,[.,.]]]
=> [1,3,2,6,5,4] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 3
[[[.,[.,.]],.],[.,[.,.]]]
=> [2,1,3,6,5,4] => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 3
[[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? = 0 + 3
[[.,[.,[.,[.,.]]]],[.,.]]
=> [4,3,2,1,6,5] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[[[.,.],[.,[.,.]]],[.,.]]
=> [1,4,3,2,6,5] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 3
[[[.,[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 0 + 3
[[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? = 0 + 3
[[[.,[.,[.,.]]],.],[.,.]]
=> [3,2,1,4,6,5] => [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 3
[[[[.,.],[.,.]],.],[.,.]]
=> [1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? = 0 + 3
[[[[.,[.,.]],.],.],[.,.]]
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ? = 0 + 3
[[[[[.,.],.],.],.],[.,.]]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? = 0 + 3
[[.,[.,[.,[.,[.,.]]]]],.]
=> [5,4,3,2,1,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,5,4,3,2,6] => [1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[[[.,[.,.]],[.,[.,.]]],.]
=> [2,1,5,4,3,6] => [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 3
[[[[.,.],.],[.,[.,.]]],.]
=> [1,2,5,4,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? = 0 + 3
[[[.,[.,[.,.]]],[.,.]],.]
=> [3,2,1,5,4,6] => [3,1,2,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001301
(load all 71 compositions to match this statistic)
(load all 71 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 23%●distinct values known / distinct values provided: 17%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 23%●distinct values known / distinct values provided: 17%
Values
[[.,.],.]
=> [.,[.,.]]
=> [2,1] => ([],2)
=> 0
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 0
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 0
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 0
[[[.,.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 0
[[[.,[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 0
[[[[.,.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[[[[.,.],[[.,.],.]],.],.],.]
=> [3,4,1,2,5,6,7] => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [[[[.,[[.,.],.]],[.,.]],.],.]
=> [5,2,3,1,4,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> [4,5,1,2,3,6,7] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[[[.,.],[[[.,.],.],.]],.],.]
=> [3,4,5,1,2,6,7] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [[[[.,.],[[.,.],[.,.]]],.],.]
=> [5,3,4,1,2,6,7] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [[[[.,.],[.,[[.,.],.]]],.],.]
=> [4,5,3,1,2,6,7] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [[[.,[[[.,.],[.,.]],.]],.],.]
=> [4,2,3,5,1,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [[[.,[[.,[[.,.],.]],.]],.],.]
=> [3,4,2,5,1,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [[[.,[[.,.],[[.,.],.]]],.],.]
=> [4,5,2,3,1,6,7] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],[.,.]],.]
=> [6,1,2,3,4,5,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 0
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [[[.,[[[.,.],.],.]],[.,.]],.]
=> [6,2,3,4,1,5,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],[[.,.],.]],.]
=> [5,6,1,2,3,4,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[[.,.],.]],.]
=> [5,6,3,1,2,4,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],[[.,.],.]],.]
=> [5,6,2,3,1,4,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [[[[.,.],.],[[[.,.],.],.]],.]
=> [4,5,6,1,2,3,7] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ? = 0
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [[[[.,.],.],[[.,.],[.,.]]],.]
=> [6,4,5,1,2,3,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [[[[.,.],.],[.,[[.,.],.]]],.]
=> [5,6,4,1,2,3,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [[[.,.],[[[[.,.],.],.],.]],.]
=> [3,4,5,6,1,2,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 0
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [[[.,.],[[[.,.],[.,.]],.]],.]
=> [5,3,4,6,1,2,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [[[.,.],[[.,[[.,.],.]],.]],.]
=> [4,5,3,6,1,2,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 0
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [[[.,.],[[[.,.],.],[.,.]]],.]
=> [6,3,4,5,1,2,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[[.,.],.]]],.]
=> [5,6,3,4,1,2,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [[[.,.],[.,[[[.,.],.],.]]],.]
=> [4,5,6,3,1,2,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [[.,[[[[[.,.],.],.],.],.]],.]
=> [2,3,4,5,6,1,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 0
[.,[[.,[[.,.],[.,[.,.]]]],.]]
=> [[.,[[[[.,.],.],[.,.]],.]],.]
=> [5,2,3,4,6,1,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [[.,[[.,[[[.,.],.],.]],.]],.]
=> [3,4,5,2,6,1,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 0
[.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> [5,6,2,3,4,1,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [[.,[[.,.],[[[.,.],.],.]]],.]
=> [4,5,6,2,3,1,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 0
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> [7,2,3,1,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 0
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [[[[[.,.],.],[.,.]],.],[.,.]]
=> [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 0
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [[[[.,.],[[.,.],.]],.],[.,.]]
=> [7,3,4,1,2,5,6] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [[[.,[[[.,.],.],.]],.],[.,.]]
=> [7,2,3,4,1,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 0
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],[.,.]],[.,.]]
=> [7,5,2,3,1,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [[[[.,.],.],[[.,.],.]],[.,.]]
=> [7,4,5,1,2,3,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [[[.,.],[[[.,.],.],.]],[.,.]]
=> [7,3,4,5,1,2,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [[[.,.],[.,[[.,.],.]]],[.,.]]
=> [7,4,5,3,1,2,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [[.,[[[.,.],[.,.]],.]],[.,.]]
=> [7,4,2,3,5,1,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [[.,[[.,[[.,.],.]],.]],[.,.]]
=> [7,3,4,2,5,1,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [[.,[[.,.],[[.,.],.]]],[.,.]]
=> [7,4,5,2,3,1,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 0
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 1
[[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],[[.,.],.]]
=> [6,7,3,1,2,4,5] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 0
Description
The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
Matching statistic: St001398
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 23%●distinct values known / distinct values provided: 17%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 23%●distinct values known / distinct values provided: 17%
Values
[[.,.],.]
=> [.,[.,.]]
=> [2,1] => ([],2)
=> 0
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 0
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 0
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 0
[[[.,.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 0
[[[.,[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 0
[[[[.,.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[[[[.,.],[[.,.],.]],.],.],.]
=> [3,4,1,2,5,6,7] => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [[[[.,[[.,.],.]],[.,.]],.],.]
=> [5,2,3,1,4,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> [4,5,1,2,3,6,7] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[[[.,.],[[[.,.],.],.]],.],.]
=> [3,4,5,1,2,6,7] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [[[[.,.],[[.,.],[.,.]]],.],.]
=> [5,3,4,1,2,6,7] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [[[[.,.],[.,[[.,.],.]]],.],.]
=> [4,5,3,1,2,6,7] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [[[.,[[[.,.],[.,.]],.]],.],.]
=> [4,2,3,5,1,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [[[.,[[.,[[.,.],.]],.]],.],.]
=> [3,4,2,5,1,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [[[.,[[.,.],[[.,.],.]]],.],.]
=> [4,5,2,3,1,6,7] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],[.,.]],.]
=> [6,1,2,3,4,5,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 0
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [[[.,[[[.,.],.],.]],[.,.]],.]
=> [6,2,3,4,1,5,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],[[.,.],.]],.]
=> [5,6,1,2,3,4,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[[.,.],.]],.]
=> [5,6,3,1,2,4,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],[[.,.],.]],.]
=> [5,6,2,3,1,4,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [[[[.,.],.],[[[.,.],.],.]],.]
=> [4,5,6,1,2,3,7] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ? = 0
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [[[[.,.],.],[[.,.],[.,.]]],.]
=> [6,4,5,1,2,3,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [[[[.,.],.],[.,[[.,.],.]]],.]
=> [5,6,4,1,2,3,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [[[.,.],[[[[.,.],.],.],.]],.]
=> [3,4,5,6,1,2,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 0
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [[[.,.],[[[.,.],[.,.]],.]],.]
=> [5,3,4,6,1,2,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [[[.,.],[[.,[[.,.],.]],.]],.]
=> [4,5,3,6,1,2,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 0
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [[[.,.],[[[.,.],.],[.,.]]],.]
=> [6,3,4,5,1,2,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[[.,.],.]]],.]
=> [5,6,3,4,1,2,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [[[.,.],[.,[[[.,.],.],.]]],.]
=> [4,5,6,3,1,2,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [[.,[[[[[.,.],.],.],.],.]],.]
=> [2,3,4,5,6,1,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 0
[.,[[.,[[.,.],[.,[.,.]]]],.]]
=> [[.,[[[[.,.],.],[.,.]],.]],.]
=> [5,2,3,4,6,1,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 0
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [[.,[[.,[[[.,.],.],.]],.]],.]
=> [3,4,5,2,6,1,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 0
[.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> [5,6,2,3,4,1,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [[.,[[.,.],[[[.,.],.],.]]],.]
=> [4,5,6,2,3,1,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 0
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> [7,2,3,1,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 0
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [[[[[.,.],.],[.,.]],.],[.,.]]
=> [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 0
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [[[[.,.],[[.,.],.]],.],[.,.]]
=> [7,3,4,1,2,5,6] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [[[.,[[[.,.],.],.]],.],[.,.]]
=> [7,2,3,4,1,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 0
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],[.,.]],[.,.]]
=> [7,5,2,3,1,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [[[[.,.],.],[[.,.],.]],[.,.]]
=> [7,4,5,1,2,3,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [[[.,.],[[[.,.],.],.]],[.,.]]
=> [7,3,4,5,1,2,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [[[.,.],[.,[[.,.],.]]],[.,.]]
=> [7,4,5,3,1,2,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 0
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [[.,[[[.,.],[.,.]],.]],[.,.]]
=> [7,4,2,3,5,1,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [[.,[[.,[[.,.],.]],.]],[.,.]]
=> [7,3,4,2,5,1,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 0
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [[.,[[.,.],[[.,.],.]]],[.,.]]
=> [7,4,5,2,3,1,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 0
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 1
[[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],[[.,.],.]]
=> [6,7,3,1,2,4,5] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 0
Description
Number of subsets of size 3 of elements in a poset that form a "v".
For a finite poset $(P,\leq)$, this is the number of sets $\{x,y,z\} \in \binom{P}{3}$ that form a "v"-subposet (i.e., a subposet consisting of a bottom element covered by two incomparable elements).
Matching statistic: St000068
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 23%●distinct values known / distinct values provided: 17%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 23%●distinct values known / distinct values provided: 17%
Values
[[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 1 + 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 0 + 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 0 + 1
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,8),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,10),(12,11),(12,13),(13,8),(13,10)],14)
=> ? = 0 + 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 0 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 0 + 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,8),(4,7),(5,2),(5,8),(6,1),(6,7),(7,10),(8,9),(10,3),(10,5)],11)
=> ? = 0 + 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> ([(0,5),(0,6),(1,8),(2,8),(4,9),(5,7),(6,4),(6,7),(7,9),(8,3),(9,1),(9,2)],10)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ([(0,5),(0,6),(1,8),(2,8),(4,9),(5,7),(6,4),(6,7),(7,9),(8,3),(9,1),(9,2)],10)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ([(0,5),(0,6),(1,9),(2,10),(3,12),(4,7),(5,2),(5,8),(6,3),(6,8),(7,9),(8,10),(8,12),(10,11),(11,1),(11,7),(12,4),(12,11)],13)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,2),(6,3),(8,9),(9,6)],10)
=> ? = 0 + 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,10),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,8),(12,11),(12,13),(13,8),(13,10)],14)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,8),(4,7),(5,2),(5,8),(6,1),(6,7),(7,10),(8,9),(10,3),(10,5)],11)
=> ? = 0 + 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
=> ? = 0 + 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> ? = 0 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
=> ? = 0 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,10),(2,9),(3,7),(4,7),(5,2),(5,8),(6,1),(6,8),(7,5),(7,6),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 0 + 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,8),(3,7),(4,9),(5,9),(6,1),(6,7),(7,8),(8,2),(9,3),(9,6)],10)
=> ? = 0 + 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
=> ? = 0 + 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
=> ? = 0 + 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ([(0,4),(0,6),(1,12),(2,9),(3,10),(4,7),(5,3),(5,8),(6,1),(6,7),(7,5),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ([(0,5),(0,6),(1,10),(3,7),(4,8),(5,9),(6,1),(6,9),(7,8),(8,2),(9,3),(9,10),(10,4),(10,7)],11)
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,10),(3,7),(4,8),(5,9),(6,1),(6,9),(7,8),(8,2),(9,3),(9,10),(10,4),(10,7)],11)
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ([(0,4),(0,6),(1,8),(3,7),(4,9),(5,2),(6,3),(6,9),(7,8),(8,5),(9,1),(9,7)],10)
=> ? = 0 + 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ([(0,4),(0,6),(1,8),(3,7),(4,9),(5,2),(6,3),(6,9),(7,8),(8,5),(9,1),(9,7)],10)
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
=> ? = 0 + 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> ? = 0 + 1
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ([(0,5),(0,6),(1,9),(2,8),(3,10),(4,11),(5,3),(5,13),(6,4),(6,13),(8,7),(9,7),(10,2),(10,12),(11,1),(11,12),(12,8),(12,9),(13,10),(13,11)],14)
=> ? = 0 + 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 0 + 1
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ([(0,5),(0,6),(1,10),(2,7),(3,7),(4,8),(5,9),(6,4),(6,9),(8,10),(9,1),(9,8),(10,2),(10,3)],11)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,8),(3,7),(4,2),(4,11),(5,4),(5,10),(6,1),(6,7),(7,5),(7,9),(9,10),(10,11),(11,8)],12)
=> ? = 0 + 1
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
=> ? = 0 + 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,8),(3,7),(4,9),(5,9),(6,1),(6,7),(7,8),(8,2),(9,3),(9,6)],10)
=> ? = 0 + 1
[.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> ? = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St001107
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
[[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 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]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 0
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 0
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 0
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 0
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 0
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 0
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 0
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 0
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 0
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 0
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 0
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 0
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 0
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 0
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 0
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 0
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 0
[.,[[[[.,[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 0
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[.,[[.,[[.,.],[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[.,[[.,[[[.,.],.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 0
[.,[[.,[[[.,.],[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 0
[.,[[.,[[[.,[.,.]],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0
[.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
[.,[[[.,[[.,.],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 0
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[.,[[[[.,[.,.]],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 0
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000678
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 18%●distinct values known / distinct values provided: 17%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 18%●distinct values known / distinct values provided: 17%
Values
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[[.,[.,.]],.],.],[.,.]]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[.,[[.,[[.,.],[.,[.,.]]]],.]]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[.,[[.,[[[.,.],.],[.,.]]],.]]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[.,[[.,[[[.,.],[.,.]],.]],.]]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[.,[[[.,.],[.,[.,[.,.]]]],.]]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001305
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001305: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 17%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001305: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 17%
Values
[[.,.],.]
=> [1,2] => [1,2] => ([],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [6,4,5,7,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [7,6,3,4,5,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [7,5,4,3,6,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [7,4,5,3,6,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [7,5,3,4,6,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [7,4,3,5,6,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [6,4,5,3,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [6,5,3,4,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [6,4,3,5,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [7,4,5,6,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [6,4,5,7,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [5,4,6,7,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [7,6,5,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [7,5,6,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [7,6,5,2,3,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [7,6,4,3,2,5,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [7,6,3,4,2,5,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
Description
The number of induced cycles on four vertices in a graph.
Matching statistic: St001306
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001306: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 17%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001306: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 17%
Values
[[.,.],.]
=> [1,2] => [1,2] => ([],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [6,4,5,7,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [7,6,3,4,5,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [7,5,4,3,6,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [7,4,5,3,6,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [7,5,3,4,6,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [7,4,3,5,6,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [6,4,5,3,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [6,5,3,4,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [6,4,3,5,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [7,4,5,6,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [6,4,5,7,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [5,4,6,7,2,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [7,6,5,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [7,5,6,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [7,6,5,2,3,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [7,6,4,3,2,5,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [7,6,3,4,2,5,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
Description
The number of induced paths on four vertices in a graph.
The following 623 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St000078The number of alternating sign matrices whose left key is the permutation. St001272The number of graphs with the same degree sequence. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000623The number of occurrences of the pattern 52341 in a permutation. St000640The rank of the largest boolean interval in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000358The number of occurrences of the pattern 31-2. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000732The number of double deficiencies of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001330The hat guessing number of a graph. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000779The tier of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St001396Number of triples of incomparable elements in a finite poset. St000210Minimum over maximum difference of elements in cycles. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St001468The smallest fixpoint of a permutation. St000666The number of right tethers of a permutation. St000485The length of the longest cycle of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000098The chromatic number of a graph. St000322The skewness of a graph. St001073The number of nowhere zero 3-flows of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001130The number of two successive successions in a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000042The number of crossings of a perfect matching. St000296The length of the symmetric border of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St000326The position of the first one in a binary word after appending a 1 at the end. St000876The number of factors in the Catalan decomposition of a binary word. St000733The row containing the largest entry of a standard tableau. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000268The number of strongly connected orientations of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000456The monochromatic index of a connected graph. St000535The rank-width of a graph. St000544The cop number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000948The chromatic discriminant of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001546The number of monomials in the Tutte polynomial of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001826The maximal number of leaves on a vertex of a graph. St001029The size of the core of a graph. St001316The domatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St000312The number of leaves in a graph. St000097The order of the largest clique of the graph. St000516The number of stretching pairs of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000845The maximal number of elements covered by an element in a poset. St000664The number of right ropes of a permutation. St001114The number of odd descents of a permutation. St001536The number of cyclic misalignments of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St001434The number of negative sum pairs of a signed permutation. St000253The crossing number of a set partition. St000627The exponent of a binary word. St000709The number of occurrences of 14-2-3 or 14-3-2. St001947The number of ties in a parking function. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000989The number of final rises of a permutation. St001381The fertility of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000041The number of nestings of a perfect matching. St001737The number of descents of type 2 in a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000355The number of occurrences of the pattern 21-3. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000872The number of very big descents of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001377The major index minus the number of inversions of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001735The number of permutations with the same set of runs. St000095The number of triangles of a graph. St000217The number of occurrences of the pattern 312 in a permutation. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000974The length of the trunk of an ordered tree. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001705The number of occurrences of the pattern 2413 in a permutation. St001871The number of triconnected components of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000570The Edelman-Greene number of a permutation. St001828The Euler characteristic of a graph. St000822The Hadwiger number of the graph. St000039The number of crossings of a permutation. St000234The number of global ascents of a permutation. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000317The cycle descent number of a permutation. St000351The determinant of the adjacency matrix of a graph. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000552The number of cut vertices of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000674The number of hills of a Dyck path. St000699The toughness times the least common multiple of 1,. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001307The number of induced stars on four vertices in a graph. St001323The independence gap of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001479The number of bridges of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001957The number of Hasse diagrams with a given underlying undirected graph. St000069The number of maximal elements of a poset. St000260The radius of a connected graph. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000313The number of degree 2 vertices of a graph. St000536The pathwidth of a graph. St000553The number of blocks of a graph. St000655The length of the minimal rise of a Dyck path. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000916The packing number of a graph. St000917The open packing number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001393The induced matching number of a graph. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001672The restrained domination number of a graph. St001765The number of connected components of the friends and strangers graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000011The number of touch points (or returns) of a Dyck path. St000258The burning number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000846The maximal number of elements covering an element of a poset. St000918The 2-limited packing number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001644The dimension of a graph. St001716The 1-improper chromatic number of a graph. St000271The chromatic index of a graph. St001108The 2-dynamic chromatic number of a graph. St000022The number of fixed points of a permutation. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000051The size of the left subtree of a binary tree. St000065The number of entries equal to -1 in an alternating sign matrix. St000090The variation of a composition. St000091The descent variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000133The "bounce" of a permutation. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000221The number of strong fixed points of a permutation. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000297The number of leading ones in a binary word. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000441The number of successions of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000461The rix statistic of a permutation. St000534The number of 2-rises of a permutation. St000629The defect of a binary word. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000731The number of double exceedences of a permutation. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001057The Grundy value of the game of creating an independent set in a graph. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001308The number of induced paths on three vertices in a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001341The number of edges in the center of a graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001394The genus of a permutation. St001429The number of negative entries in a signed permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001703The villainy of a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000079The number of alternating sign matrices for a given Dyck path. St000092The number of outer peaks of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000099The number of valleys of a permutation, including the boundary. St000115The single entry in the last row. St000137The Grundy value of an integer partition. St000259The diameter of a connected graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000349The number of different adjacency matrices of a graph. St000382The first part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000392The length of the longest run of ones in a binary word. St000618The number of self-evacuating tableaux of given shape. St000654The first descent of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000669The number of permutations obtained by switching ascents or descents of size 2. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000756The sum of the positions of the left to right maxima of a permutation. St000763The sum of the positions of the strong records of an integer composition. St000764The number of strong records in an integer composition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000805The number of peaks of the associated bargraph. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000884The number of isolated descents of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000897The number of different multiplicities of parts of an integer partition. St000905The number of different multiplicities of parts of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000991The number of right-to-left minima of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001282The number of graphs with the same chromatic polynomial. St001286The annihilation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001386The number of prime labellings of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001463The number of distinct columns in the nullspace of a graph. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001512The minimum rank of a graph. St001642The Prague dimension of a graph. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001777The number of weak descents in an integer composition. St001812The biclique partition number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001931The weak major index of an integer composition regarded as a word. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000007The number of saliances of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000834The number of right outer peaks of a permutation. St000842The breadth of a permutation. St001093The detour number of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000219The number of occurrences of the pattern 231 in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001811The Castelnuovo-Mumford regularity of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000045The number of linear extensions of a binary tree. St000647The number of big descents of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000862The number of parts of the shifted shape of a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001060The distinguishing index of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000058The order of a permutation. St001260The permanent of an alternating sign matrix. St001116The game chromatic number of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000741The Colin de Verdière graph invariant. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001545The second Elser number of a connected graph. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000717The number of ordinal summands of a poset. St001964The interval resolution global dimension of a poset. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000302The determinant of the distance matrix of a connected graph. St000363The number of minimal vertex covers of a graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000181The number of connected components of the Hasse diagram for the poset. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001827The number of two-component spanning forests of a graph. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000117The number of centered tunnels of a Dyck path. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000241The number of cyclical small excedances. St000295The length of the border of a binary word. St000478Another weight of a partition according to Alladi. St000488The number of cycles of a permutation of length at most 2. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000895The number of ones on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St000084The number of subtrees. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000667The greatest common divisor of the parts of the partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000843The decomposition number of a perfect matching. St000993The multiplicity of the largest part of an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001729The number of visible descents of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St001110The 3-dynamic chromatic number of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001845The number of join irreducibles minus the rank of a lattice. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001846The number of elements which do not have a complement in the lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001866The nesting alignments of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001783The number of odd automorphisms of a graph. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001118The acyclic chromatic index of a graph.
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