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Your data matches 529 different statistics following compositions of up to 3 maps.
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Matching statistic: St000222
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 2
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 2
Description
The number of alignments in the permutation.
Matching statistic: St001822
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001822: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001822: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 0
[3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 1
[2,3,4,1] => [2,3,4,1] => 0
[2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [2,4,3,1] => 1
[3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 1
[3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 2
Description
The number of alignments of a signed permutation.
An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, [[St001866]], an alignment of type EN, [[St001867]], or an alignment of type NE, [[St001868]].
Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, [[St001862]], let \operatorname{wex} be the number of weak excedances, [[St001863]], and let \operatorname{neg} be the number of negative entries, [[St001429]]. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
Matching statistic: St000021
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => 0
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [3,2,1] => 2
[2,3,4,1] => [2,3,1] => [3,2,1] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [3,2,1] => 2
[3,1,2,4] => [3,1,2] => [3,2,1] => 2
[3,1,4,2] => [3,1,2] => [3,2,1] => 2
[3,2,1,4] => [3,2,1] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => [3,2,1] => 2
[3,4,2,1] => [3,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [3,2,1] => 2
[4,3,1,2] => [3,1,2] => [3,2,1] => 2
[4,3,2,1] => [3,2,1] => [3,2,1] => 2
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000214
Mp00252: Permutations —restriction⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => 0
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [3,2,1] => 2
[2,3,4,1] => [2,3,1] => [3,2,1] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [3,2,1] => 2
[3,1,2,4] => [3,1,2] => [3,2,1] => 2
[3,1,4,2] => [3,1,2] => [3,2,1] => 2
[3,2,1,4] => [3,2,1] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => [3,2,1] => 2
[3,4,2,1] => [3,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [3,2,1] => 2
[4,3,1,2] => [3,1,2] => [3,2,1] => 2
[4,3,2,1] => [3,2,1] => [3,2,1] => 2
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000246
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => 0
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [2,1] => 0
[2,3,1] => [2,1] => [2,1] => 0
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => [1,3,2] => 2
[1,2,4,3] => [1,2,3] => [1,3,2] => 2
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,3,2] => 2
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [2,1,3] => 2
[2,1,4,3] => [2,1,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => [2,3,1] => 1
[2,3,4,1] => [2,3,1] => [2,3,1] => 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 2
[2,4,3,1] => [2,3,1] => [2,3,1] => 1
[3,1,2,4] => [3,1,2] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 0
[3,2,4,1] => [3,2,1] => [3,2,1] => 0
[3,4,1,2] => [3,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 0
[4,1,2,3] => [1,2,3] => [1,3,2] => 2
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [2,1,3] => 2
[4,2,3,1] => [2,3,1] => [2,3,1] => 1
[4,3,1,2] => [3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 0
Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000662
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => 0
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [3,2,1] => 2
[2,3,4,1] => [2,3,1] => [3,2,1] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [3,2,1] => 2
[3,1,2,4] => [3,1,2] => [3,2,1] => 2
[3,1,4,2] => [3,1,2] => [3,2,1] => 2
[3,2,1,4] => [3,2,1] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => [3,2,1] => 2
[3,4,2,1] => [3,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [3,2,1] => 2
[4,3,1,2] => [3,1,2] => [3,2,1] => 2
[4,3,2,1] => [3,2,1] => [3,2,1] => 2
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St001723
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [1,1] => ([(0,1)],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 0
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 0
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 0
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 0
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 0
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The differential of a graph.
The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$.
The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$.
The differential of a graph is the maximal differential of a set of vertices.
Matching statistic: St001724
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001724: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001724: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [1,1] => ([(0,1)],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 0
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 0
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 0
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 0
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 0
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The 2-packing differential of a graph.
The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$.
The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$.
A set $S\subseteq V(G)$ is $2$-packing if the closed neighbourhoods of any two vertices in $S$ have empty intersection.
The $2$-packing differential of a graph is the maximal differential of any $2$-packing set of vertices.
Matching statistic: St001727
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 0
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 0
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => 1
[1,4,2,3] => [1,3,4,2] => [2,4,1,3] => 1
[1,4,3,2] => [1,4,3,2] => [3,4,2,1] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 1
[2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 2
[2,4,1,3] => [4,3,1,2] => [1,4,3,2] => 1
[2,4,3,1] => [2,4,3,1] => [4,2,3,1] => 2
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1
[3,1,4,2] => [2,3,4,1] => [4,1,2,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [3,2,4,1] => [4,2,1,3] => 2
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 2
[3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 2
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 2
[4,1,3,2] => [2,4,3,1] => [4,2,3,1] => 2
[4,2,1,3] => [3,2,4,1] => [4,2,1,3] => 2
[4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 2
[4,3,1,2] => [3,4,2,1] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Matching statistic: St000047
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [2] => 1 = 0 + 1
[2,1] => [2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [1,3,2] => [2,1] => 2 = 1 + 1
[1,3,2] => [1,3,2] => [2,1] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,2] => 1 = 0 + 1
[2,3,1] => [2,3,1] => [2,1] => 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [1,4,3,2] => [2,1,1] => 3 = 2 + 1
[1,2,4,3] => [1,4,3,2] => [2,1,1] => 3 = 2 + 1
[1,3,2,4] => [1,4,3,2] => [2,1,1] => 3 = 2 + 1
[1,3,4,2] => [1,4,3,2] => [2,1,1] => 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [2,1,1] => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,1] => 3 = 2 + 1
[2,1,3,4] => [2,1,4,3] => [1,2,1] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [2,2] => 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => [2,1,1] => 3 = 2 + 1
[2,4,1,3] => [2,4,1,3] => [2,2] => 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => [2,1,1] => 3 = 2 + 1
[3,1,2,4] => [3,1,4,2] => [1,2,1] => 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [1,2,1] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => 1 = 0 + 1
[3,2,4,1] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [2,2] => 3 = 2 + 1
[3,4,2,1] => [3,4,2,1] => [2,1,1] => 3 = 2 + 1
[4,1,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [1,1,2] => 1 = 0 + 1
[4,2,3,1] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => [1,1,2] => 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
Description
The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
The following 519 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000883The number of longest increasing subsequences of a permutation. St000004The major index of a permutation. St000009The charge of a standard tableau. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000141The maximum drop size of a permutation. St000157The number of descents of a standard tableau. St000209Maximum difference of elements in cycles. St000218The number of occurrences of the pattern 213 in a permutation. St000225Difference between largest and smallest parts in a partition. St000237The number of small exceedances. St000245The number of ascents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000356The number of occurrences of the pattern 13-2. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000441The number of successions of a permutation. St000463The number of admissible inversions of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000546The number of global descents of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000682The Grundy value of Welter's game on a binary word. St000703The number of deficiencies of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000921The number of internal inversions of a binary word. St000996The number of exclusive left-to-right maxima of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001090The number of pop-stack-sorts needed to sort a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. 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. St001234The number of indecomposable three dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001438The number of missing boxes of a skew partition. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001565The number of arithmetic progressions of length 2 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. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001638The book thickness of a graph. St001644The dimension of a graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001684The reduced word complexity of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001695The natural comajor index of a standard Young tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St000058The order of a permutation. St000071The number of maximal chains in a poset. St000110The number of permutations less than or equal to a permutation in left weak order. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000381The largest part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000638The number of up-down runs of a permutation. St000808The number of up steps of the associated bargraph. St000935The number of ordered refinements of an integer partition. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001066The number of simple reflexive modules in the corresponding 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001497The position of the largest weak excedence of a permutation. St000240The number of indices that are not small excedances. St000156The Denert index of a permutation. St000171The degree of the graph. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000304The load of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000651The maximal size of a rise in a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001489The maximum of the number of descents and the number of inverse descents. St001557The number of inversions of the second entry of a permutation. St001671Haglund's hag of a permutation. St001726The number of visible inversions of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001861The number of Bruhat lower covers of a permutation. St001869The maximum cut size of a graph. St001894The depth of a signed permutation. St000060The greater neighbor of the maximum. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000626The minimal period of a binary word. St000722The number of different neighbourhoods in a graph. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001268The size of the largest ordinal summand in the poset. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001725The harmonious chromatic number of a graph. St001779The order of promotion on the set of linear extensions of a poset. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000067The inversion number of the alternating sign matrix. St000081The number of edges of a graph. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length of a permutation. St000272The treewidth of a graph. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000305The inverse major index of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000369The dinv deficit of a Dyck path. St000446The disorder of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000497The lcb statistic of a set partition. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000538The number of even inversions of a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000572The dimension exponent of a set partition. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000632The jump number of the poset. St000653The last descent of a permutation. St000691The number of changes of a binary word. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000836The number of descents of distance 2 of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000868The aid statistic in the sense of Shareshian-Wachs. St000931The number of occurrences of the pattern UUU in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001117The game chromatic index of a graph. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001298The number of repeated entries in the Lehmer code of a permutation. St001358The largest degree of a regular subgraph of a graph. St001375The pancake length of a permutation. St001388The number of non-attacking neighbors of a permutation. St001427The number of descents of a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001479The number of bridges of a graph. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001512The minimum rank of a graph. St001569The maximal modular displacement of a permutation. St001649The length of a longest trail in a graph. St001721The degree of a binary word. St001743The discrepancy of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001792The arboricity of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001812The biclique partition number of a graph. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001826The maximal number of leaves on a vertex of a graph. St001896The number of right descents of a signed permutations. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001960The number of descents of a permutation minus one if its first entry is not one. St001962The proper pathwidth of a graph. St000006The dinv of a Dyck path. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000086The number of subgraphs. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000485The length of the longest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000740The last entry of a permutation. St000822The Hadwiger number of the graph. St000839The largest opener of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St000909The number of maximal chains of maximal size in a poset. St000983The length of the longest alternating subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001081The number of minimal length factorizations of a permutation into star transpositions. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001128The exponens consonantiae of a partition. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001399The distinguishing number of a poset. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000064The number of one-box pattern of a permutation. 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. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000327The number of cover relations in a poset. St000354The number of recoils of a permutation. St000503The maximal difference between two elements in a common block. St000539The number of odd inversions of a permutation. St000728The dimension of a set partition. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001061The number of indices that are both descents and recoils of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001480The number of simple summands of the module J^2/J^3. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000288The number of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000693The modular (standard) major index of a standard tableau. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St001077The prefix exchange distance of a permutation. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001372The length of a longest cyclic run of ones of a binary word. St001592The maximal number of simple paths between any two different vertices of a graph. St000444The length of the maximal rise of a Dyck path. St000489The number of cycles of a permutation of length at most 3. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000910The number of maximal chains of minimal length in a poset. St000933The number of multipartitions of sizes given by an integer partition. St000990The first ascent of a permutation. St001346The number of parking functions that give the same permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001959The product of the heights of the peaks of a Dyck path. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000259The diameter of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001877Number of indecomposable injective modules with projective dimension 2. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001570The minimal number of edges to add to make a graph Hamiltonian. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001118The acyclic chromatic index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000937The number of positive values of the symmetric group character corresponding to the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001624The breadth of a lattice. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000744The length of the path to the largest entry in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000456The monochromatic index of a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001645The pebbling number of a connected graph. St001651The Frankl number of a lattice. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001933The largest multiplicity of a part in an integer partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000667The greatest common divisor of the parts of the partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001571The Cartan determinant of the integer partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001568The smallest positive integer that does not appear twice in the partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000420The number of Dyck paths that are weakly above a Dyck path. St000478Another weight of a partition according to Alladi. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001808The box weight or horizontal decoration of a Dyck path. St001875The number of simple modules with projective dimension at most 1. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001857The number of edges in the reduced word graph of a signed permutation. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000422The energy of a graph, if it is integral. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000284The Plancherel distribution on integer partitions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000736The last entry in the first row of a semistandard tableau. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001435The number of missing boxes in the first row. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001488The number of corners of a skew partition. St000782The indicator function of whether a given perfect matching is an L & P matching. 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 $eA$. 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 $eA$. 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 $eA$. St001722The number of minimal chains with small intervals between a binary word and the top element. St001407The number of minimal entries in a semistandard tableau. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000302The determinant of the distance matrix of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000928The sum of the coefficients of the character polynomial 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. St000997The even-odd crank of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000102The charge of a semistandard tableau. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000274The number of perfect matchings of a graph. 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. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001625The Möbius invariant of a lattice. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. 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. St001783The number of odd automorphisms of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001856The number of edges in the reduced word graph of a permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. 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. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000739The first entry in the last row of a semistandard tableau. 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 $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001828The Euler characteristic of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St001734The lettericity of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph.
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