searching the database
Your data matches 910 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000092
(load all 230 compositions to match this statistic)
(load all 230 compositions to match this statistic)
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 1
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 2
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000099
(load all 258 compositions to match this statistic)
(load all 258 compositions to match this statistic)
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 2
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000862
(load all 278 compositions to match this statistic)
(load all 278 compositions to match this statistic)
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 2
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Matching statistic: St000023
(load all 258 compositions to match this statistic)
(load all 258 compositions to match this statistic)
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 0 = 1 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 1 = 2 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 0 = 1 - 1
[3,1,4,2] => 1 = 2 - 1
[3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 1 = 2 - 1
[3,4,2,1] => 1 = 2 - 1
[4,1,2,3] => 0 = 1 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => 0 = 1 - 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000201
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 1
[1,2] => [.,[.,.]]
=> 1
[2,1] => [[.,.],.]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> 2
[3,2,1] => [[[.,.],.],.]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> 1
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000325
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 2
[2,1,3] => [1,3,2] => 2
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,2,3] => 1
[3,2,1] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => 1
[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] => 2
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,2,3] => 2
[2,1,3,4] => [1,3,4,2] => 2
[2,1,4,3] => [1,4,2,3] => 2
[2,3,1,4] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,3,2,4] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,2,4,3] => 2
[3,1,4,2] => [1,4,2,3] => 2
[3,2,1,4] => [1,4,2,3] => 2
[3,2,4,1] => [1,2,4,3] => 2
[3,4,1,2] => [1,2,3,4] => 1
[3,4,2,1] => [1,2,3,4] => 1
[4,1,2,3] => [1,2,3,4] => 1
[4,1,3,2] => [1,3,2,4] => 2
[4,2,1,3] => [1,3,2,4] => 2
[4,2,3,1] => [1,2,3,4] => 1
[4,3,1,2] => [1,2,3,4] => 1
[4,3,2,1] => [1,2,3,4] => 1
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000396
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 1
[1,2] => [.,[.,.]]
=> 1
[2,1] => [[.,.],.]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> 2
[3,2,1] => [[[.,.],.],.]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> 1
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000470
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 2
[2,1,3] => [1,3,2] => 2
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,2,3] => 1
[3,2,1] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => 1
[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] => 2
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,2,3] => 2
[2,1,3,4] => [1,3,4,2] => 2
[2,1,4,3] => [1,4,2,3] => 2
[2,3,1,4] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,3,2,4] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,2,4,3] => 2
[3,1,4,2] => [1,4,2,3] => 2
[3,2,1,4] => [1,4,2,3] => 2
[3,2,4,1] => [1,2,4,3] => 2
[3,4,1,2] => [1,2,3,4] => 1
[3,4,2,1] => [1,2,3,4] => 1
[4,1,2,3] => [1,2,3,4] => 1
[4,1,3,2] => [1,3,2,4] => 2
[4,2,1,3] => [1,3,2,4] => 2
[4,2,3,1] => [1,2,3,4] => 1
[4,3,1,2] => [1,2,3,4] => 1
[4,3,2,1] => [1,2,3,4] => 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000758
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [2] => 1
[2,1] => [1,1] => 1
[1,2,3] => [3] => 1
[1,3,2] => [2,1] => 1
[2,1,3] => [1,2] => 2
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 2
[3,2,1] => [1,1,1] => 1
[1,2,3,4] => [4] => 1
[1,2,4,3] => [3,1] => 1
[1,3,2,4] => [2,2] => 2
[1,3,4,2] => [3,1] => 1
[1,4,2,3] => [2,2] => 2
[1,4,3,2] => [2,1,1] => 1
[2,1,3,4] => [1,3] => 2
[2,1,4,3] => [1,2,1] => 2
[2,3,1,4] => [2,2] => 2
[2,3,4,1] => [3,1] => 1
[2,4,1,3] => [2,2] => 2
[2,4,3,1] => [2,1,1] => 1
[3,1,2,4] => [1,3] => 2
[3,1,4,2] => [1,2,1] => 2
[3,2,1,4] => [1,1,2] => 2
[3,2,4,1] => [1,2,1] => 2
[3,4,1,2] => [2,2] => 2
[3,4,2,1] => [2,1,1] => 1
[4,1,2,3] => [1,3] => 2
[4,1,3,2] => [1,2,1] => 2
[4,2,1,3] => [1,1,2] => 2
[4,2,3,1] => [1,2,1] => 2
[4,3,1,2] => [1,1,2] => 2
[4,3,2,1] => [1,1,1,1] => 1
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000920
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
The following 900 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000662The staircase size of the code of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St000010The length of the partition. St000054The first entry of the permutation. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000308The height of the tree associated to a permutation. 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. St000390The number of runs of ones in a binary word. St000397The Strahler number of a rooted tree. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000542The number of left-to-right-minima of a permutation. St000679The pruning number of an ordered tree. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000822The Hadwiger number of the graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001029The size of the core of a graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001116The game chromatic number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. 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. St001432The order dimension of the partition. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001484The number of singletons of an integer partition. St001486The number of corners of the ribbon associated with an integer composition. St001487The number of inner corners of a skew partition. St001494The Alon-Tarsi number of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001597The Frobenius rank of a skew partition. St001670The connected partition number of a graph. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001883The mutual visibility number of a graph. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001963The tree-depth of a graph. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000142The number of even parts of a partition. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. 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. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000336The leg major index of a standard tableau. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000360The number of occurrences of the pattern 32-1. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000523The number of 2-protected nodes of a rooted tree. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000759The smallest missing part in an integer partition. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000996The number of exclusive left-to-right maxima of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001176The size of a partition minus its first part. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001394The genus of a permutation. St001427The number of descents of a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001644The dimension of a graph. St001657The number of twos in an integer partition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001726The number of visible inversions of a permutation. St001728The number of invisible descents of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001743The discrepancy of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001823The Stasinski-Voll length of a signed permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001896The number of right descents of a signed permutations. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001931The weak major index of an integer composition regarded as a word. St001946The number of descents in a parking function. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000053The number of valleys of the Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000069The number of maximal elements of a poset. St000086The number of subgraphs. St000087The number of induced subgraphs. St000108The number of partitions contained in the given partition. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000183The side length of the Durfee square of an integer partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000273The domination number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000306The bounce count of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000328The maximum number of child nodes in a tree. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000343The number of spanning subgraphs of a graph. St000354The number of recoils of a permutation. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000389The number of runs of ones of odd length in a binary word. St000409The number of pitchforks in a binary tree. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000479The Ramsey number of a graph. St000507The number of ascents of a standard tableau. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000522The number of 1-protected nodes of a rooted tree. St000528The height of a poset. St000532The total number of rook placements on a Ferrers board. St000544The cop number of a graph. St000553The number of blocks of a graph. St000636The hull number of a graph. St000659The number of rises of length at least 2 of a Dyck path. St000669The number of permutations obtained by switching ascents or descents of size 2. St000675The number of centered multitunnels of a Dyck path. St000676The number of odd rises of a Dyck path. St000701The protection number of a binary tree. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000829The Ulam distance of a permutation to the identity permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St000971The smallest closer of a set partition. St000972The composition number of a graph. St000991The number of right-to-left minima of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001058The breadth of the ordered tree. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001096The size of the overlap set of a permutation. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001224Let X be the direct sum of all simple modules 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. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001286The annihilation number of a graph. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001352The number of internal nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001471The magnitude of a Dyck path. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001488The number of corners of a skew partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001555The order of a signed permutation. St001569The maximal modular displacement of a permutation. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001716The 1-improper chromatic number of a graph. St001717The largest size of an interval in a poset. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. 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. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001802The number of endomorphisms of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001814The number of partitions interlacing the given partition. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001884The number of borders of a binary word. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000004The major index of a permutation. St000009The charge of a standard tableau. St000012The area of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000024The number of double up and double down steps of a Dyck path. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000080The rank of the poset. St000081The number of edges of a graph. St000091The descent variation of a composition. St000094The depth of an ordered tree. St000117The number of centered tunnels of a Dyck path. St000148The number of odd parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000156The Denert index of a permutation. St000160The multiplicity of the smallest part of a partition. St000168The number of internal nodes of an ordered tree. St000169The cocharge of a standard tableau. St000171The degree of the graph. St000211The rank of the set partition. St000228The size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000274The number of perfect matchings of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000330The (standard) major index of a standard tableau. St000340The number of non-final maximal constant sub-paths of length greater than one. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000361The second Zagreb index of a graph. St000367The number of simsun double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000439The position of the first down step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000459The hook length of the base cell of a partition. St000466The Gutman (or modified Schultz) index of a connected graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000521The number of distinct subtrees of an ordered tree. St000547The number of even non-empty partial sums of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000552The number of cut vertices of a graph. St000628The balance of a binary word. St000651The maximal size of a rise in a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000663The number of right floats of a permutation. St000665The number of rafts of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000678The number of up steps after the last double rise of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000769The major index of a composition regarded as a word. St000778The metric dimension of a graph. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000845The maximal number of elements covered by an element in a poset. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000871The number of very big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000891The number of distinct diagonal sums of a permutation matrix. St000897The number of different multiplicities of parts of an integer partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001127The sum of the squares of the parts of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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 $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001323The independence gap of a graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001391The disjunction number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001429The number of negative entries in a signed permutation. St001459The number of zero columns in the nullspace of a graph. St001470The cyclic holeyness of a permutation. St001479The number of bridges of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001512The minimum rank of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001638The book thickness of a graph. St001649The length of a longest trail in a graph. St001673The degree of asymmetry of an integer composition. 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. St001692The number of vertices with higher degree than the average degree in a graph. St001695The natural comajor index of a standard Young tableau. St001697The shifted natural comajor index of a standard Young tableau. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001712The number of natural descents of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001742The difference of the maximal and the minimal degree in a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001769The reflection length of a signed permutation. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001842The major index of a set partition. St001864The number of excedances of a signed permutation. St001869The maximum cut size of a graph. St001895The oddness of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001935The number of ascents in a parking function. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001949The rigidity index of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000353The number of inner valleys of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000539The number of odd inversions of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000485The length of the longest cycle of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000640The rank of the largest boolean interval in a poset. St000652The maximal difference between successive positions of a permutation. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000886The number of permutations with the same antidiagonal sums. St000933The number of multipartitions of sizes given by an integer partition. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001128The exponens consonantiae of a partition. St001246The maximal difference between two consecutive entries of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000216The absolute length of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000392The length of the longest run of ones in a binary word. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. 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. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000779The tier of a permutation. St000809The reduced reflection length of the permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000893The number of distinct diagonal sums of an alternating sign matrix. St000919The number of maximal left branches of a binary tree. 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. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001372The length of a longest cyclic run of ones of a binary word. St001413Half the length of the longest even length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001424The number of distinct squares in a binary word. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001592The maximal number of simple paths between any two different vertices of a graph. St000061The number of nodes on the left branch of a binary tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000744The length of the path to the largest entry in a standard Young tableau. 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. St000946The sum of the skew hook positions in a Dyck path. St000983The length of the longest alternating subword. St000990The first ascent of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001052The length of the exterior of a permutation. St001062The maximal size of a block of a set partition. St001220The width of a permutation. St001267The length of the Lyndon factorization of the binary word. St001346The number of parking functions that give the same permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001531Number of partial orders contained in the poset determined by the Dyck path. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000347The inversion sum of a binary word. St000391The sum of the positions of the ones in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000489The number of cycles of a permutation of length at most 3. St000538The number of even inversions of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000565The major index of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. 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. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (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. 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. St000653The last descent of a permutation. St000658The number of rises of length 2 of a Dyck path. St000674The number of hills 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. St000691The number of changes of a binary word. St000693The modular (standard) major index of a standard tableau. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000732The number of double deficiencies of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000792The Grundy value for the game of ruler on a binary word. St000794The mak of a permutation. St000836The number of descents of distance 2 of a permutation. St000961The shifted major index of a permutation. St000984The number of boxes below precisely one peak. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001114The number of odd descents of a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001153The number of blocks with even minimum in a set partition. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001480The number of simple summands of the module J^2/J^3. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001557The number of inversions of the second entry of a permutation. St001668The number of points of the poset minus the width of the poset. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001820The size of the image of the pop stack sorting operator. St000677The standardized bi-alternating inversion number of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000526The number of posets with combinatorially isomorphic order polytopes. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000455The second largest eigenvalue of a graph if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000219The number of occurrences of the pattern 231 in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000144The pyramid weight of the Dyck path. St000184The size of the centralizer of any permutation of given cycle type. St000460The hook length of the last cell along the main diagonal of an integer partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001118The acyclic chromatic index of a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000681The Grundy value of Chomp on Ferrers diagrams. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000944The 3-degree of an integer partition. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. 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. St001964The interval resolution global dimension of a poset. 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. St000302The determinant of the distance matrix of a connected graph. St000422The energy of a graph, if it is integral. St000467The hyper-Wiener index of a connected graph. St000661The number of rises of length 3 of a Dyck path. St001933The largest multiplicity of a part in an integer partition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000100The number of linear extensions of a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000633The size of the automorphism group of a poset. St000667The greatest common divisor of the parts of the partition. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001890The maximum magnitude of the Möbius function of a poset. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. 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. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St000420The number of Dyck paths that are weakly above a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001272The number of graphs with the same degree sequence. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001808The box weight or horizontal decoration of a Dyck path. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St000805The number of peaks of the associated bargraph. 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. St000900The minimal number of repetitions of a part in an integer composition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001481The minimal height of a peak of a Dyck path. St001518The number of graphs with the same ordinary spectrum as the given graph. 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. St001722The number of minimal chains with small intervals between a binary word and the top element. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. 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. St000322The skewness of a graph. St000338The number of pixed points of a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000906The length of the shortest maximal chain in a poset. St001130The number of two successive successions in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001705The number of occurrences of the pattern 2413 in a permutation. St001871The number of triconnected components of a graph. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a 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. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001877Number of indecomposable injective modules with projective dimension 2. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000567The sum of the products of all pairs of parts. 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. St000929The constant term of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. 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. St000102The charge of a semistandard tableau. St000309The number of vertices with even degree. 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. St000736The last entry in the first row of a semistandard tableau. 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. St001613The binary logarithm of the size of the center 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. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. 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. St001845The number of join irreducibles minus the rank of a lattice. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!