- FindStat

Your data matches 308 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000530
(load all 105 compositions to match this statistic)

Mapping

St000530: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 1 = 0 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 2 = 1 + 1
[3,1,2] => 2 = 1 + 1
[3,2,1] => 1 = 0 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 3 = 2 + 1
[1,3,4,2] => 3 = 2 + 1
[1,4,3,2] => 3 = 2 + 1
[2,1,3,4] => 3 = 2 + 1
[2,3,4,1] => 3 = 2 + 1
[3,2,1,4] => 3 = 2 + 1
[3,4,2,1] => 3 = 2 + 1
[4,1,2,3] => 3 = 2 + 1
[4,2,1,3] => 3 = 2 + 1
[4,3,1,2] => 3 = 2 + 1
[4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => 1 = 0 + 1
[5,4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5,6] => 1 = 0 + 1
[6,5,4,3,2,1] => 1 = 0 + 1

Description

The number of permutations with the same descent word as the given permutation. The descent word of a permutation is the binary word given by [[Mp00109]]. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map [[Mp00109]] containing a given permutation. This statistic appears as ''up-down analysis'' in statistical applications in genetics, see [1] and the references therein.

Matching statistic: St001958
(load all 1047 compositions to match this statistic)

Mapping

St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 1 = 0 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 2 = 1 + 1
[3,1,2] => 2 = 1 + 1
[3,2,1] => 1 = 0 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 3 = 2 + 1
[1,3,4,2] => 3 = 2 + 1
[1,4,3,2] => 3 = 2 + 1
[2,1,3,4] => 3 = 2 + 1
[2,3,4,1] => 3 = 2 + 1
[3,2,1,4] => 3 = 2 + 1
[3,4,2,1] => 3 = 2 + 1
[4,1,2,3] => 3 = 2 + 1
[4,2,1,3] => 3 = 2 + 1
[4,3,1,2] => 3 = 2 + 1
[4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => 1 = 0 + 1
[5,4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5,6] => 1 = 0 + 1
[6,5,4,3,2,1] => 1 = 0 + 1

Description

The degree of the polynomial interpolating the values of a permutation. Given a permutation

$\pi\in\mathfrak S_n$ there is a polynomial

$p$ of minimal degree such that

$p(n)=\pi(n)$ for

$n\in\{1,\dots,n\}$ . This statistic records the degree of

$p$ .

Matching statistic: St000003
(load all 13 compositions to match this statistic)

Mapping

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [2]
 => 1 = 0 + 1
[2,1] => [1,1]
 => 1 = 0 + 1
[1,2,3] => [3]
 => 1 = 0 + 1
[1,3,2] => [2,1]
 => 2 = 1 + 1
[2,1,3] => [2,1]
 => 2 = 1 + 1
[2,3,1] => [2,1]
 => 2 = 1 + 1
[3,1,2] => [2,1]
 => 2 = 1 + 1
[3,2,1] => [1,1,1]
 => 1 = 0 + 1
[1,2,3,4] => [4]
 => 1 = 0 + 1
[1,2,4,3] => [3,1]
 => 3 = 2 + 1
[1,3,4,2] => [3,1]
 => 3 = 2 + 1
[1,4,3,2] => [2,1,1]
 => 3 = 2 + 1
[2,1,3,4] => [3,1]
 => 3 = 2 + 1
[2,3,4,1] => [3,1]
 => 3 = 2 + 1
[3,2,1,4] => [2,1,1]
 => 3 = 2 + 1
[3,4,2,1] => [2,1,1]
 => 3 = 2 + 1
[4,1,2,3] => [3,1]
 => 3 = 2 + 1
[4,2,1,3] => [2,1,1]
 => 3 = 2 + 1
[4,3,1,2] => [2,1,1]
 => 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
 => 1 = 0 + 1
[1,2,3,4,5] => [5]
 => 1 = 0 + 1
[5,4,3,2,1] => [1,1,1,1,1]
 => 1 = 0 + 1
[1,2,3,4,5,6] => [6]
 => 1 = 0 + 1
[6,5,4,3,2,1] => [1,1,1,1,1,1]
 => 1 = 0 + 1

Description

The number of [[/StandardTableaux|standard Young tableaux]] of the partition.

Matching statistic: St000075
(load all 122 compositions to match this statistic)

Mapping

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000075: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [[1,2]]
 => 1 = 0 + 1
[2,1] => [[1],[2]]
 => 1 = 0 + 1
[1,2,3] => [[1,2,3]]
 => 1 = 0 + 1
[1,3,2] => [[1,2],[3]]
 => 2 = 1 + 1
[2,1,3] => [[1,3],[2]]
 => 2 = 1 + 1
[2,3,1] => [[1,2],[3]]
 => 2 = 1 + 1
[3,1,2] => [[1,3],[2]]
 => 2 = 1 + 1
[3,2,1] => [[1],[2],[3]]
 => 1 = 0 + 1
[1,2,3,4] => [[1,2,3,4]]
 => 1 = 0 + 1
[1,2,4,3] => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,3,4,2] => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,4,3,2] => [[1,2],[3],[4]]
 => 3 = 2 + 1
[2,1,3,4] => [[1,3,4],[2]]
 => 3 = 2 + 1
[2,3,4,1] => [[1,2,3],[4]]
 => 3 = 2 + 1
[3,2,1,4] => [[1,4],[2],[3]]
 => 3 = 2 + 1
[3,4,2,1] => [[1,2],[3],[4]]
 => 3 = 2 + 1
[4,1,2,3] => [[1,3,4],[2]]
 => 3 = 2 + 1
[4,2,1,3] => [[1,4],[2],[3]]
 => 3 = 2 + 1
[4,3,1,2] => [[1,4],[2],[3]]
 => 3 = 2 + 1
[4,3,2,1] => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[1,2,3,4,5] => [[1,2,3,4,5]]
 => 1 = 0 + 1
[5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
 => 1 = 0 + 1
[6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
 => 1 = 0 + 1

Description

The orbit size of a standard tableau under promotion.

Matching statistic: St000277
(load all 11 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [2] => 1 = 0 + 1
[2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [3] => 1 = 0 + 1
[1,3,2] => [2,1] => 2 = 1 + 1
[2,1,3] => [1,2] => 2 = 1 + 1
[2,3,1] => [2,1] => 2 = 1 + 1
[3,1,2] => [1,2] => 2 = 1 + 1
[3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [4] => 1 = 0 + 1
[1,2,4,3] => [3,1] => 3 = 2 + 1
[1,3,4,2] => [3,1] => 3 = 2 + 1
[1,4,3,2] => [2,1,1] => 3 = 2 + 1
[2,1,3,4] => [1,3] => 3 = 2 + 1
[2,3,4,1] => [3,1] => 3 = 2 + 1
[3,2,1,4] => [1,1,2] => 3 = 2 + 1
[3,4,2,1] => [2,1,1] => 3 = 2 + 1
[4,1,2,3] => [1,3] => 3 = 2 + 1
[4,2,1,3] => [1,1,2] => 3 = 2 + 1
[4,3,1,2] => [1,1,2] => 3 = 2 + 1
[4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => [5] => 1 = 0 + 1
[5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5,6] => [6] => 1 = 0 + 1
[6,5,4,3,2,1] => [1,1,1,1,1,1] => 1 = 0 + 1

Description

The number of ribbon shaped standard tableaux. A ribbon is a connected skew shape which does not contain a

$2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape. This is also the size of the preimage of the map 'descent composition' [[Mp00071]] from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed. For a composition

$c=c_1,\dots,c_k$ of

$n$ , the number of ribbon shaped standard tableaux equals

$\sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell},$ where the sum is over all coarsenings of

$c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]

Matching statistic: St000529
(load all 16 compositions to match this statistic)

Mapping

Mp00109: Permutations —descent word⟶ Binary words
St000529: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => 1 = 0 + 1
[2,1] => 1 => 1 = 0 + 1
[1,2,3] => 00 => 1 = 0 + 1
[1,3,2] => 01 => 2 = 1 + 1
[2,1,3] => 10 => 2 = 1 + 1
[2,3,1] => 01 => 2 = 1 + 1
[3,1,2] => 10 => 2 = 1 + 1
[3,2,1] => 11 => 1 = 0 + 1
[1,2,3,4] => 000 => 1 = 0 + 1
[1,2,4,3] => 001 => 3 = 2 + 1
[1,3,4,2] => 001 => 3 = 2 + 1
[1,4,3,2] => 011 => 3 = 2 + 1
[2,1,3,4] => 100 => 3 = 2 + 1
[2,3,4,1] => 001 => 3 = 2 + 1
[3,2,1,4] => 110 => 3 = 2 + 1
[3,4,2,1] => 011 => 3 = 2 + 1
[4,1,2,3] => 100 => 3 = 2 + 1
[4,2,1,3] => 110 => 3 = 2 + 1
[4,3,1,2] => 110 => 3 = 2 + 1
[4,3,2,1] => 111 => 1 = 0 + 1
[1,2,3,4,5] => 0000 => 1 = 0 + 1
[5,4,3,2,1] => 1111 => 1 = 0 + 1
[1,2,3,4,5,6] => 00000 => 1 = 0 + 1
[6,5,4,3,2,1] => 11111 => 1 = 0 + 1

Description

The number of permutations whose descent word is the given binary word. This is the sizes of the preimages of the map [[Mp00109]].

Matching statistic: St000543
(load all 426 compositions to match this statistic)

Mapping

Mp00130: Permutations —descent tops⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => 1 = 0 + 1
[2,1] => 1 => 1 = 0 + 1
[1,2,3] => 00 => 1 = 0 + 1
[1,3,2] => 01 => 2 = 1 + 1
[2,1,3] => 10 => 2 = 1 + 1
[2,3,1] => 01 => 2 = 1 + 1
[3,1,2] => 01 => 2 = 1 + 1
[3,2,1] => 11 => 1 = 0 + 1
[1,2,3,4] => 000 => 1 = 0 + 1
[1,2,4,3] => 001 => 3 = 2 + 1
[1,3,4,2] => 001 => 3 = 2 + 1
[1,4,3,2] => 011 => 3 = 2 + 1
[2,1,3,4] => 100 => 3 = 2 + 1
[2,3,4,1] => 001 => 3 = 2 + 1
[3,2,1,4] => 110 => 3 = 2 + 1
[3,4,2,1] => 101 => 3 = 2 + 1
[4,1,2,3] => 001 => 3 = 2 + 1
[4,2,1,3] => 101 => 3 = 2 + 1
[4,3,1,2] => 011 => 3 = 2 + 1
[4,3,2,1] => 111 => 1 = 0 + 1
[1,2,3,4,5] => 0000 => 1 = 0 + 1
[5,4,3,2,1] => 1111 => 1 = 0 + 1
[1,2,3,4,5,6] => 00000 => 1 = 0 + 1
[6,5,4,3,2,1] => 11111 => 1 = 0 + 1

Description

The size of the conjugacy class of a binary word. Two words

$u$ and

$v$ are conjugate, if

$u=w_1 w_2$ and

$v=w_2 w_1$ , see Section 1.3 of [1].

Matching statistic: St000626
(load all 16 compositions to match this statistic)

Mapping

Mp00109: Permutations —descent word⟶ Binary words
St000626: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => 1 = 0 + 1
[2,1] => 1 => 1 = 0 + 1
[1,2,3] => 00 => 1 = 0 + 1
[1,3,2] => 01 => 2 = 1 + 1
[2,1,3] => 10 => 2 = 1 + 1
[2,3,1] => 01 => 2 = 1 + 1
[3,1,2] => 10 => 2 = 1 + 1
[3,2,1] => 11 => 1 = 0 + 1
[1,2,3,4] => 000 => 1 = 0 + 1
[1,2,4,3] => 001 => 3 = 2 + 1
[1,3,4,2] => 001 => 3 = 2 + 1
[1,4,3,2] => 011 => 3 = 2 + 1
[2,1,3,4] => 100 => 3 = 2 + 1
[2,3,4,1] => 001 => 3 = 2 + 1
[3,2,1,4] => 110 => 3 = 2 + 1
[3,4,2,1] => 011 => 3 = 2 + 1
[4,1,2,3] => 100 => 3 = 2 + 1
[4,2,1,3] => 110 => 3 = 2 + 1
[4,3,1,2] => 110 => 3 = 2 + 1
[4,3,2,1] => 111 => 1 = 0 + 1
[1,2,3,4,5] => 0000 => 1 = 0 + 1
[5,4,3,2,1] => 1111 => 1 = 0 + 1
[1,2,3,4,5,6] => 00000 => 1 = 0 + 1
[6,5,4,3,2,1] => 11111 => 1 = 0 + 1

Description

The minimal period of a binary word. This is the smallest natural number

$p$ such that

$w_i=w_{i+p}$ for all

$i\in\{1,\dots,|w|-p\}$ .

Matching statistic: St000704
(load all 17 compositions to match this statistic)

Mapping

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [2]
 => 1 = 0 + 1
[2,1] => [1,1]
 => 1 = 0 + 1
[1,2,3] => [3]
 => 1 = 0 + 1
[1,3,2] => [2,1]
 => 2 = 1 + 1
[2,1,3] => [2,1]
 => 2 = 1 + 1
[2,3,1] => [2,1]
 => 2 = 1 + 1
[3,1,2] => [2,1]
 => 2 = 1 + 1
[3,2,1] => [1,1,1]
 => 1 = 0 + 1
[1,2,3,4] => [4]
 => 1 = 0 + 1
[1,2,4,3] => [3,1]
 => 3 = 2 + 1
[1,3,4,2] => [3,1]
 => 3 = 2 + 1
[1,4,3,2] => [2,1,1]
 => 3 = 2 + 1
[2,1,3,4] => [3,1]
 => 3 = 2 + 1
[2,3,4,1] => [3,1]
 => 3 = 2 + 1
[3,2,1,4] => [2,1,1]
 => 3 = 2 + 1
[3,4,2,1] => [2,1,1]
 => 3 = 2 + 1
[4,1,2,3] => [3,1]
 => 3 = 2 + 1
[4,2,1,3] => [2,1,1]
 => 3 = 2 + 1
[4,3,1,2] => [2,1,1]
 => 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
 => 1 = 0 + 1
[1,2,3,4,5] => [5]
 => 1 = 0 + 1
[5,4,3,2,1] => [1,1,1,1,1]
 => 1 = 0 + 1
[1,2,3,4,5,6] => [6]
 => 1 = 0 + 1
[6,5,4,3,2,1] => [1,1,1,1,1,1]
 => 1 = 0 + 1

Description

The number of semistandard tableaux on a given integer partition with minimal maximal entry. This is, for an integer partition

$\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$ , the number of [[SemistandardTableaux|semistandard tableaux]] of shape

$\lambda$ with maximal entry

$k$ . Equivalently, this is the evaluation

$s_\lambda(1,\ldots,1)$ of the Schur function

$s_\lambda$ in

$k$ variables, or, explicitly,

$\prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$ where the product is over all cells

$(i,j) \in L$ and

$\operatorname{hook}(i,j)$ is the hook length of a cell. See [Theorem 6.3, 1] for details.

Matching statistic: St001780
(load all 13 compositions to match this statistic)

Mapping

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [2]
 => 1 = 0 + 1
[2,1] => [1,1]
 => 1 = 0 + 1
[1,2,3] => [3]
 => 1 = 0 + 1
[1,3,2] => [2,1]
 => 2 = 1 + 1
[2,1,3] => [2,1]
 => 2 = 1 + 1
[2,3,1] => [2,1]
 => 2 = 1 + 1
[3,1,2] => [2,1]
 => 2 = 1 + 1
[3,2,1] => [1,1,1]
 => 1 = 0 + 1
[1,2,3,4] => [4]
 => 1 = 0 + 1
[1,2,4,3] => [3,1]
 => 3 = 2 + 1
[1,3,4,2] => [3,1]
 => 3 = 2 + 1
[1,4,3,2] => [2,1,1]
 => 3 = 2 + 1
[2,1,3,4] => [3,1]
 => 3 = 2 + 1
[2,3,4,1] => [3,1]
 => 3 = 2 + 1
[3,2,1,4] => [2,1,1]
 => 3 = 2 + 1
[3,4,2,1] => [2,1,1]
 => 3 = 2 + 1
[4,1,2,3] => [3,1]
 => 3 = 2 + 1
[4,2,1,3] => [2,1,1]
 => 3 = 2 + 1
[4,3,1,2] => [2,1,1]
 => 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
 => 1 = 0 + 1
[1,2,3,4,5] => [5]
 => 1 = 0 + 1
[5,4,3,2,1] => [1,1,1,1,1]
 => 1 = 0 + 1
[1,2,3,4,5,6] => [6]
 => 1 = 0 + 1
[6,5,4,3,2,1] => [1,1,1,1,1,1]
 => 1 = 0 + 1

Description

The order of promotion on the set of standard tableaux of given shape.

The following 298 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000057The Shynar inversion number of a standard tableau. St000222The number of alignments in the permutation. St000293The number of inversions of a binary word. St000369The dinv deficit of a Dyck path. St000387The matching number of a graph. St000682The Grundy value of Welter's game on a binary word. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001910The height of the middle non-run of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000071The number of maximal chains in a poset. St000172The Grundy number of a graph. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. St001415The length of the longest palindromic prefix of a binary word. St001510The number of self-evacuating linear extensions of a finite poset. St001581The achromatic number of a graph. St001595The number of standard Young tableaux of the skew partition. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001963The tree-depth of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000089The absolute variation of a composition. St000185The weighted size of a partition. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000290The major index of a binary word. St000355The number of occurrences of the pattern 21-3. St000362The size of a minimal vertex cover of a graph. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000747A variant of the major index of a set partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001176The size of a partition minus its first part. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001695The natural comajor index of a standard Young tableau. St001812The biclique partition number of a graph. St001961The sum of the greatest common divisors of all pairs of parts. St000001The number of reduced words for a permutation. St000014The number of parking functions supported by a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000048The multinomial of the parts of a partition. St000078The number of alternating sign matrices whose left key is the permutation. St000100The number of linear extensions of a poset. St000255The number of reduced Kogan faces with the permutation as type. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000452The number of distinct eigenvalues of a graph. St000548The number of different non-empty partial sums of an integer partition. St000670The reversal length of a permutation. 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. St000988The orbit size of a permutation under Foata's bijection. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001304The number of maximally independent sets of vertices of a graph. St001512The minimum rank of a graph. St001725The harmonious chromatic number of a graph. St001779The order of promotion on the set of linear extensions of a poset. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000058The order of a permutation. St001093The detour 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. St001883The mutual visibility number of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000209Maximum difference of elements in cycles. St000494The number of inversions of distance at most 3 of a permutation. St000789The number of crossing-similar perfect matchings of a perfect matching. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000485The length of the longest cycle of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001117The game chromatic index of a graph. 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. St001742The difference of the maximal and the minimal degree in a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000171The degree of the graph. St000219The number of occurrences of the pattern 231 in a permutation. St000271The chromatic index of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000567The sum of the products of all pairs of parts. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. 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. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the 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. St001349The number of different graphs obtained from the given graph by removing an edge. St001381The fertility of a permutation. St001438The number of missing boxes of a skew partition. St001535The number of cyclic alignments of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000029The depth of a permutation. St000045The number of linear extensions of a binary tree. St000216The absolute length of a permutation. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000420The number of Dyck paths that are weakly above a Dyck path. St000456The monochromatic index of a connected graph. St000537The cutwidth of a graph. St001110The 3-dynamic chromatic number of a graph. 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. St001271The competition number of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001480The number of simple summands of the module J^2/J^3. St000141The maximum drop size of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000881The number of short braid edges in the graph of braid moves of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001727The number of invisible inversions of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000454The largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St001644The dimension of a graph. St001330The hat guessing number of 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. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000259The diameter of a connected graph. St001118The acyclic chromatic index of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001623The number of doubly irreducible elements of a lattice. St001875The number of simple modules with projective dimension at most 1. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000552The number of cut vertices of a graph. St000768The number of peaks in an integer composition. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001470The cyclic holeyness of a permutation. St001793The difference between the clique number and the chromatic number of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000785The number of distinct colouring schemes of a graph. St001282The number of graphs with the same chromatic polynomial. St001333The cardinality of a minimal edge-isolating set of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000098The chromatic number of a graph. St000166The depth minus 1 of an ordered tree. 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. St000636The hull number of a graph. St000808The number of up steps of the associated bargraph. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001654The monophonic hull number of a 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001964The interval resolution global dimension of a poset. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St000095The number of triangles of a graph. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St000822The Hadwiger number of the graph. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001488The number of corners of a skew partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St000553The number of blocks of a graph. St001391The disjunction number of a graph. St000422The energy of a graph, if it is integral. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000181The number of connected components of the Hasse diagram for the poset. St000741The Colin de Verdière graph invariant. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001890The maximum magnitude of the Möbius function of a poset. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000509The diagonal index (content) of a partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001568The smallest positive integer that does not appear twice in the partition.