- FindStat

Your data matches 262 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000021
(load all 69 compositions to match this statistic)

Mapping

St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Matching statistic: St000157
(load all 23 compositions to match this statistic)

Mapping

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Matching statistic: St000245
(load all 76 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of a permutation.

Matching statistic: St000662
(load all 16 compositions to match this statistic)

Mapping

Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The staircase size of the code of a permutation. The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ . The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ . This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.

Matching statistic: St000703
(load all 58 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of deficiencies of a permutation. This is defined as

$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$ The number of exceedances is [[St000155]].

Matching statistic: St000377

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv defect of an integer partition. This is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$ .

Matching statistic: St000053
(load all 25 compositions to match this statistic)

Mapping

Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] =>  => [1] => [1,0]
 => 0
[1,2] => 0 => [2] => [1,1,0,0]
 => 0
[2,1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2,3] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[2,1,3] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3,4] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,3,2,4] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,4,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,4,2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,4,3,2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,3,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,3,1,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,3,4,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,3,1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[3,1,2,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[3,2,1,4] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[3,2,4,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[3,4,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,4,2,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,2,1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,2,3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,3,1,2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[4,3,2,1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,3,4,5] => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,2,4,3,5] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,4,5,3] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,5,3,4] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,5,4,3] => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,3,2,4,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,2,5,4] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,4,2,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,4,5,2] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,5,2,4] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,5,4,2] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,4,2,3,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,4,2,5,3] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,3,2,5] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,3,5,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,5,2,3] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1

Description

The number of valleys of the Dyck path.

Matching statistic: St000272
(load all 15 compositions to match this statistic)

Mapping

Mp00130: Permutations —descent tops⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] =>  => [1] => ([],1)
 => 0
[1,2] => 0 => [2] => ([],2)
 => 0
[2,1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,2,3] => 00 => [3] => ([],3)
 => 0
[1,3,2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,3] => 10 => [1,2] => ([(1,2)],3)
 => 1
[2,3,1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,1,2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => 000 => [4] => ([],4)
 => 0
[1,2,4,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,3,2,4] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,3,4,2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,4,2,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => 100 => [1,3] => ([(2,3)],4)
 => 1
[2,1,4,3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,3,4,1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,1,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,3,1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,1,4,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,4,1,2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,4,2,1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,3,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,1,3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,1,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => 0000 => [5] => ([],5)
 => 0
[1,2,3,5,4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,3,5] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,5,3] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,3,2,5,4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,2,5] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,3,4,5,2] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,2,4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,4,2] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,4,2,5,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1

Description

The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.

Matching statistic: St000306
(load all 8 compositions to match this statistic)

Mapping

Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] =>  => [1] => [1,0]
 => 0
[1,2] => 0 => [2] => [1,1,0,0]
 => 0
[2,1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2,3] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[2,1,3] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3,4] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,3,2,4] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,4,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,4,2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,4,3,2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,3,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,3,1,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,3,4,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,3,1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[3,1,2,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[3,2,1,4] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[3,2,4,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[3,4,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,4,2,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,2,1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,2,3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[4,3,1,2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[4,3,2,1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,3,4,5] => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,2,4,3,5] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,4,5,3] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,5,3,4] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,5,4,3] => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,3,2,4,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,2,5,4] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,4,2,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,4,5,2] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,5,2,4] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,5,4,2] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,4,2,3,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,4,2,5,3] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,3,2,5] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,3,5,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,5,2,3] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

Matching statistic: St000362
(load all 14 compositions to match this statistic)

Mapping

Mp00130: Permutations —descent tops⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] =>  => [1] => ([],1)
 => 0
[1,2] => 0 => [2] => ([],2)
 => 0
[2,1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,2,3] => 00 => [3] => ([],3)
 => 0
[1,3,2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,3] => 10 => [1,2] => ([(1,2)],3)
 => 1
[2,3,1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,1,2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => 000 => [4] => ([],4)
 => 0
[1,2,4,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,3,2,4] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,3,4,2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,4,2,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => 100 => [1,3] => ([(2,3)],4)
 => 1
[2,1,4,3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,3,4,1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,1,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,3,1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,1,4,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,4,1,2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,4,2,1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,3,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,1,3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,1,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => 0000 => [5] => ([],5)
 => 0
[1,2,3,5,4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,3,5] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,5,3] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,3,2,5,4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,2,5] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,3,4,5,2] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,2,4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,4,2] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,4,2,5,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1

Description

The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . Finding a minimal vertex cover is an NP-hard optimization problem.

The following 252 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000441The number of successions of a permutation. St000536The pathwidth of a graph. St000672The number of minimal elements in Bruhat order not less than the permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001777The number of weak descents in an integer composition. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call 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. 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: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000155The number of exceedances (also excedences) of a permutation. St000024The number of double up and double down steps of a Dyck path. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001176The size of a partition minus its first part. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000507The number of ascents of a standard tableau. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000074The number of special entries. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000171The degree of the graph. St000224The sorting index of a permutation. St000228The size of a partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000310The minimal degree of a vertex of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000369The dinv deficit of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000454The largest eigenvalue of a graph if it is integral. St000546The number of global descents of a permutation. St000632The jump number of the poset. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001215Let 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001270The bandwidth of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. 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. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001726The number of visible inversions of a permutation. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000084The number of subtrees. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000164The number of short pairs. St000201The number of leaf nodes in a binary tree. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000390The number of runs of ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000469The distinguishing number of a graph. St000527The width of the poset. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000843The decomposition number of a perfect matching. St000912The number of maximal antichains in a poset. St000926The clique-coclique number of a graph. St000991The number of right-to-left minima of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. 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. 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. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number 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. St000439The position of the first down step of a Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra 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. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000354The number of recoils of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000702The number of weak deficiencies of a permutation. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000728The dimension of a set partition. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001372The length of a longest cyclic run of ones of a binary word. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St000061The number of nodes on the left branch of a binary tree. St000444The length of the maximal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000925The number of topologically connected components of a set partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000159The number of distinct parts of the integer partition. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001427The number of descents of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001727The number of invisible inversions of a permutation. St001668The number of points of the poset minus the width of the poset. St001651The Frankl number of a lattice. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001946The number of descents in a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001712The number of natural descents of a standard Young tableau. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. 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

$eA$ . St001624The breadth of a lattice.