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Your data matches 436 different statistics following compositions of up to 3 maps.
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Matching statistic: St001804
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St001804: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1
[[1,2]]
=> 1
[[1],[2]]
=> 1
[[1,2,3]]
=> 1
[[1,3],[2]]
=> 2
[[1,2],[3]]
=> 1
[[1],[2],[3]]
=> 1
[[1,2,3,4]]
=> 1
[[1,3,4],[2]]
=> 2
[[1,2,4],[3]]
=> 2
[[1,2,3],[4]]
=> 1
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 1
[[1,4],[2],[3]]
=> 3
[[1,3],[2],[4]]
=> 2
[[1,2],[3],[4]]
=> 2
[[1],[2],[3],[4]]
=> 1
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St000358
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0 = 1 - 1
[[1,2]]
=> [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => 0 = 1 - 1
[[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => 1 = 2 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001744
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001744: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0 = 1 - 1
[[1,2]]
=> [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => 0 = 1 - 1
[[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => 1 = 2 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St000531
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Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> []
=> 1
[[1,2]]
=> [2]
=> []
=> 1
[[1],[2]]
=> [1,1]
=> [1]
=> 1
[[1,2,3]]
=> [3]
=> []
=> 1
[[1,3],[2]]
=> [2,1]
=> [1]
=> 1
[[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,3,4]]
=> [4]
=> []
=> 1
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 2
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
Description
The leading coefficient of the rook polynomial of an integer partition.
Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.
Matching statistic: St001659
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Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001659: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001659: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> []
=> 1
[[1,2]]
=> [2]
=> []
=> 1
[[1],[2]]
=> [1,1]
=> [1]
=> 1
[[1,2,3]]
=> [3]
=> []
=> 1
[[1,3],[2]]
=> [2,1]
=> [1]
=> 1
[[1,2],[3]]
=> [2,1]
=> [1]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,3,4]]
=> [4]
=> []
=> 1
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> 1
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> 1
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 2
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
Description
The number of ways to place as many non-attacking rooks as possible on a Ferrers board.
Matching statistic: St000039
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 0 = 1 - 1
[[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000204
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Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [.,.]
=> 0 = 1 - 1
[[1,2]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> 0 = 1 - 1
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 0 = 1 - 1
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000223
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 0 = 1 - 1
[[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,3,2] => 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,4,3,2] => 1 = 2 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,4,3] => 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,4,2] => 0 = 1 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => 1 = 2 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
Description
The number of nestings in the permutation.
Matching statistic: St000317
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 0 = 1 - 1
[[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => 0 = 1 - 1
Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Matching statistic: St000355
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(load all 5 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 0 = 1 - 1
[[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => 0 = 1 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 0 = 1 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
The following 426 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000356The number of occurrences of the pattern 13-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000534The number of 2-rises of a permutation. St000647The number of big descents of a permutation. St000663The number of right floats of a permutation. St000731The number of double exceedences of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001214The aft of an integer partition. St001323The independence gap of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001435The number of missing boxes in the first row. St001565The number of arithmetic progressions of length 2 in a permutation. St001638The book thickness of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001727The number of invisible inversions of a permutation. St000058The order of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000255The number of reduced Kogan faces with the permutation as type. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000325The width of the tree associated to a permutation. St000346The number of coarsenings of a partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000638The number of up-down runs of a permutation. St000816The number of standard composition tableaux of the composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. 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. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001642The Prague dimension of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001814The number of partitions interlacing the given partition. St001896The number of right descents of a signed permutations. 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. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000002The number of occurrences of the pattern 123 in a permutation. St000010The length of the partition. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents 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. St000052The number of valleys of a Dyck path not on the x-axis. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000155The number of exceedances (also excedences) of a permutation. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000177The number of free tiles in the pattern. St000178Number of free entries. St000209Maximum difference of elements in cycles. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000222The number of alignments in the permutation. St000225Difference between largest and smallest parts in a partition. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000245The number of ascents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000339The maf index of a permutation. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000441The number of successions of a permutation. St000446The disorder of a permutation. St000463The number of admissible inversions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000548The number of different non-empty partial sums of an integer partition. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. 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. St000742The number of big ascents of a permutation after prepending zero. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000766The number of inversions of an integer composition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000921The number of internal inversions of a binary word. St000996The number of exclusive left-to-right maxima of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand 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. St001411The number of patterns 321 or 3412 in a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001489The maximum of the number of descents and the number of inverse descents. 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. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001684The reduced word complexity of a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001726The number of visible inversions of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001931The weak major index of an integer composition regarded as a word. St000886The number of permutations with the same antidiagonal sums. St000988The orbit size of a permutation under Foata's bijection. St001052The length of the exterior of a permutation. St001220The width of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. 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. St001960The number of descents of a permutation minus one if its first entry is not one. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000652The maximal difference between successive positions of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000982The length of the longest constant subword. St001081The number of minimal length factorizations of a permutation into star transpositions. St001246The maximal difference between two consecutive entries of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001820The size of the image of the pop stack sorting operator. St001884The number of borders of a binary word. St000064The number of one-box pattern of a permutation. St000216The absolute length of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000354The number of recoils of a permutation. St000369The dinv deficit of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000491The number of inversions of a set partition. 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. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000651The maximal size of a rise in a permutation. St000795The mad of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001298The number of repeated entries in the Lehmer code of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000489The number of cycles of a permutation of length at most 3. St000527The width of the poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. 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. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001498The normalised height of a Nakayama algebra with magnitude 1. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000228The size of a partition. St000335The difference of lower and upper interactions. St000384The maximal part of the shifted composition of an integer partition. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000459The hook length of the base cell of a partition. St000519The largest length of a factor maximising the subword complexity. St000784The maximum of the length and the largest part of the integer partition. St000922The minimal number such that all substrings of this length are unique. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. 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. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. 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. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001571The Cartan determinant of the integer partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001959The product of the heights of the peaks of a Dyck path. St001964The interval resolution global dimension of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001487The number of inner corners of a skew partition. St001779The order of promotion on the set of linear extensions of a poset. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001438The number of missing boxes of a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001527The cyclic permutation representation number of an integer partition. St000422The energy of a graph, if it is integral. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. 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. St001432The order dimension of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000100The number of linear extensions of a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the 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. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001568The smallest positive integer that does not appear twice in the partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001857The number of edges in the reduced word graph of a signed permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000420The number of Dyck paths that are weakly above a Dyck path. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000667The greatest common divisor of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St001118The acyclic chromatic index of a graph. St001378The product of the cohook lengths of the integer partition. St001389The number of partitions of the same length below the given integer partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. 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. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001808The box weight or horizontal decoration of a Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001564The value of the forgotten symmetric functions when all variables set to 1. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a 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. St001924The number of cells in an integer partition whose arm and leg length coincide. 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. St000464The Schultz index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001060The distinguishing index of a graph. St001545The second Elser number of a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000181The number of connected components of the Hasse diagram for the poset. St000284The Plancherel distribution on integer partitions. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000514The number of invariant simple graphs 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000997The even-odd crank of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. 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 $eA$. St000102The charge of a semistandard tableau.
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