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Your data matches 112 different statistics following compositions of up to 3 maps.
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Matching statistic: St000929
(load all 78 compositions to match this statistic)
(load all 78 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2]
=> 0
[[.,.],.]
=> [1,2] => [1,1]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 0
[[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,1,1]
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,1,1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,1,1]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [4,1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,1,1]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,1,1,1]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,2]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,2,1]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [2,1,1,1]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,2,1]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,2,1]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [2,1,1,1]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001139
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 1
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0
Description
The number of occurrences of hills of size 2 in a Dyck path.
A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.
Matching statistic: St000268
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000268: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000268: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => ([],2)
=> ([],2)
=> 1
[[.,.],.]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> ([],3)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
Description
The number of strongly connected orientations of a graph.
Matching statistic: St000297
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => 0 => 0 => 0
[[.,.],.]
=> [1,2] => 1 => 1 => 1
[.,[.,[.,.]]]
=> [3,2,1] => 00 => 01 => 0
[.,[[.,.],.]]
=> [2,3,1] => 00 => 01 => 0
[[.,.],[.,.]]
=> [3,1,2] => 00 => 01 => 0
[[.,[.,.]],.]
=> [2,1,3] => 01 => 00 => 0
[[[.,.],.],.]
=> [1,2,3] => 11 => 10 => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 000 => 010 => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 000 => 010 => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 000 => 010 => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 000 => 010 => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 000 => 010 => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 000 => 010 => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 000 => 010 => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 000 => 010 => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 000 => 010 => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 001 => 011 => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 001 => 011 => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 001 => 011 => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 011 => 001 => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => 111 => 101 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0000 => 0101 => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0000 => 0101 => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 0000 => 0101 => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0000 => 0101 => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0000 => 0101 => 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 0000 => 0101 => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0000 => 0101 => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0000 => 0101 => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 0000 => 0101 => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0000 => 0101 => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0000 => 0101 => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 0000 => 0101 => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0000 => 0101 => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0000 => 0101 => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0000 => 0101 => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0000 => 0101 => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0000 => 0101 => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 0000 => 0101 => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0000 => 0101 => 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0000 => 0101 => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 0000 => 0101 => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0000 => 0101 => 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0000 => 0101 => 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0000 => 0101 => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 0000 => 0101 => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0000 => 0101 => 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0000 => 0101 => 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0000 => 0101 => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0001 => 0100 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000344
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000344: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000344: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => ([],2)
=> ([],2)
=> 1
[[.,.],.]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> ([],3)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
Description
The number of strongly connected outdegree sequences of a graph.
This is the evaluation of the Tutte polynomial at $x=0$ and $y=1$. According to [1,2], the set of strongly connected outdegree sequences is in bijection with strongly connected minimal orientations and also with external spanning trees.
Matching statistic: St000658
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[[.,.],.]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0
Description
The number of rises of length 2 of a Dyck path.
This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1].
A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St001073
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001073: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001073: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => ([],2)
=> ([],2)
=> 1
[[.,.],.]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> ([],3)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
Description
The number of nowhere zero 3-flows of a graph.
This is the absolute value of the evaluation of the Tutte polynomial of the graph at $(x,y)=(0,-2)$.
For $4$-regular graphs, this coincides with the number of Eulerian orientations.
Matching statistic: St001125
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
Description
The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.
Matching statistic: St001477
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001477: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001477: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => ([],2)
=> ([],2)
=> 1
[[.,.],.]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> ([],3)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
Description
The number of nowhere zero 5-flows of a graph.
Matching statistic: St001478
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001478: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001478: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => ([],2)
=> ([],2)
=> 1
[[.,.],.]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> ([],3)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
Description
The number of nowhere zero 4-flows of a graph.
The following 102 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001618The cardinality of the Frattini sublattice of a lattice. St001657The number of twos in an integer partition. St000025The number of initial rises of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000439The position of the first down step of a Dyck path. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St000944The 3-degree of an integer partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001826The maximal number of leaves on a vertex of a graph. St000884The number of isolated descents of a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000629The defect of a binary word. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000214The number of adjacencies of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000546The number of global descents of a permutation. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000234The number of global ascents of a permutation. St000296The length of the symmetric border of a binary word. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000455The second largest eigenvalue of a graph if it is integral. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001371The length of the longest Yamanouchi prefix of a binary word. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001479The number of bridges of a graph. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001498The normalised height of a Nakayama algebra with magnitude 1. St000990The first ascent of a permutation. 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. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000315The number of isolated vertices of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001367The smallest number which does not occur as degree of a vertex in a graph. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001691The number of kings in a graph. St000056The decomposition (or block) number of a permutation. St000654The first descent of a permutation. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001481The minimal height of a peak of a Dyck path. St000842The breadth of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000124The cardinality of the preimage of the Simion-Schmidt map. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000302The determinant of the distance matrix of a connected graph. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001948The number of augmented double ascents of a permutation. 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 $c_{n−1}$. St001851The number of Hecke atoms of a signed permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001462The number of factors of a standard tableaux under concatenation.
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