Your data matches 576 different statistics following compositions of up to 3 maps.
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Matching statistic: St001359
(load all 158 compositions to match this statistic)
Mapping
St001359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 = 2 - 1
[1,2] => 1 = 2 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 1 = 2 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 2 = 3 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 1 = 2 - 1
[1,2,3,4] => 1 = 2 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => 2 = 3 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 2 = 3 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 2 = 3 - 1
[3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => 2 = 3 - 1
[3,4,1,2] => 1 = 2 - 1
[3,4,2,1] => 2 = 3 - 1
[4,1,2,3] => 2 = 3 - 1
[4,1,3,2] => 2 = 3 - 1
[4,2,1,3] => 2 = 3 - 1
[4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => 2 = 3 - 1
[4,3,2,1] => 1 = 2 - 1
Description
The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. In other words, this is $2^k$ where $k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition. The generating function for the number of equivalence classes, $f(n)$, is $$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$$
Matching statistic: St000092
(load all 171 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,2] => [1,2] => 1 = 2 - 1
[2,1] => [2,1] => 1 = 2 - 1
[1,2,3] => [1,3,2] => 1 = 2 - 1
[1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => 2 = 3 - 1
[2,3,1] => [2,3,1] => 1 = 2 - 1
[3,1,2] => [3,1,2] => 2 = 3 - 1
[3,2,1] => [3,2,1] => 1 = 2 - 1
[1,2,3,4] => [1,4,3,2] => 1 = 2 - 1
[1,2,4,3] => [1,4,3,2] => 1 = 2 - 1
[1,3,2,4] => [1,4,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,4,3] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => [2,4,1,3] => 2 = 3 - 1
[2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => [3,1,4,2] => 2 = 3 - 1
[3,1,4,2] => [3,1,4,2] => 2 = 3 - 1
[3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[3,2,4,1] => [3,2,4,1] => 2 = 3 - 1
[3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[4,1,2,3] => [4,1,3,2] => 2 = 3 - 1
[4,1,3,2] => [4,1,3,2] => 2 = 3 - 1
[4,2,1,3] => [4,2,1,3] => 2 = 3 - 1
[4,2,3,1] => [4,2,3,1] => 2 = 3 - 1
[4,3,1,2] => [4,3,1,2] => 2 = 3 - 1
[4,3,2,1] => [4,3,2,1] => 1 = 2 - 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: St000660
(load all 27 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 0 = 2 - 2
[1,2] => [1,0,1,0]
 => 0 = 2 - 2
[2,1] => [1,1,0,0]
 => 0 = 2 - 2
[1,2,3] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[1,3,2] => [1,0,1,1,0,0]
 => 0 = 2 - 2
[2,1,3] => [1,1,0,0,1,0]
 => 0 = 2 - 2
[2,3,1] => [1,1,0,1,0,0]
 => 0 = 2 - 2
[3,1,2] => [1,1,1,0,0,0]
 => 1 = 3 - 2
[3,2,1] => [1,1,1,0,0,0]
 => 1 = 3 - 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 0 = 2 - 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 0 = 2 - 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 0 = 2 - 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 0 = 2 - 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 0 = 2 - 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 0 = 2 - 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0 = 2 - 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 0 = 2 - 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 0 = 2 - 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1 = 3 - 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1 = 3 - 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1 = 3 - 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1 = 3 - 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
Description
The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000884
(load all 69 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [1,2] => 0 = 2 - 2
[2,1] => [1,2] => 0 = 2 - 2
[1,2,3] => [1,2,3] => 0 = 2 - 2
[1,3,2] => [1,2,3] => 0 = 2 - 2
[2,1,3] => [1,2,3] => 0 = 2 - 2
[2,3,1] => [1,2,3] => 0 = 2 - 2
[3,1,2] => [1,3,2] => 1 = 3 - 2
[3,2,1] => [1,3,2] => 1 = 3 - 2
[1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,2,4,3] => [1,2,3,4] => 0 = 2 - 2
[1,3,2,4] => [1,2,3,4] => 0 = 2 - 2
[1,3,4,2] => [1,2,3,4] => 0 = 2 - 2
[1,4,2,3] => [1,2,4,3] => 1 = 3 - 2
[1,4,3,2] => [1,2,4,3] => 1 = 3 - 2
[2,1,3,4] => [1,2,3,4] => 0 = 2 - 2
[2,1,4,3] => [1,2,3,4] => 0 = 2 - 2
[2,3,1,4] => [1,2,3,4] => 0 = 2 - 2
[2,3,4,1] => [1,2,3,4] => 0 = 2 - 2
[2,4,1,3] => [1,2,4,3] => 1 = 3 - 2
[2,4,3,1] => [1,2,4,3] => 1 = 3 - 2
[3,1,2,4] => [1,3,2,4] => 1 = 3 - 2
[3,1,4,2] => [1,3,4,2] => 1 = 3 - 2
[3,2,1,4] => [1,3,2,4] => 1 = 3 - 2
[3,2,4,1] => [1,3,4,2] => 1 = 3 - 2
[3,4,1,2] => [1,3,2,4] => 1 = 3 - 2
[3,4,2,1] => [1,3,2,4] => 1 = 3 - 2
[4,1,2,3] => [1,4,3,2] => 0 = 2 - 2
[4,1,3,2] => [1,4,2,3] => 1 = 3 - 2
[4,2,1,3] => [1,4,3,2] => 0 = 2 - 2
[4,2,3,1] => [1,4,2,3] => 1 = 3 - 2
[4,3,1,2] => [1,4,2,3] => 1 = 3 - 2
[4,3,2,1] => [1,4,2,3] => 1 = 3 - 2
Description
The number of isolated descents of a permutation. A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001086
(load all 76 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [1,2] => 0 = 2 - 2
[2,1] => [2,1] => 0 = 2 - 2
[1,2,3] => [1,2,3] => 0 = 2 - 2
[1,3,2] => [1,3,2] => 1 = 3 - 2
[2,1,3] => [2,1,3] => 0 = 2 - 2
[2,3,1] => [1,3,2] => 1 = 3 - 2
[3,1,2] => [3,1,2] => 0 = 2 - 2
[3,2,1] => [3,2,1] => 0 = 2 - 2
[1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,2,4,3] => [1,2,4,3] => 1 = 3 - 2
[1,3,2,4] => [1,3,2,4] => 1 = 3 - 2
[1,3,4,2] => [1,2,4,3] => 1 = 3 - 2
[1,4,2,3] => [1,4,2,3] => 1 = 3 - 2
[1,4,3,2] => [1,4,3,2] => 1 = 3 - 2
[2,1,3,4] => [2,1,3,4] => 0 = 2 - 2
[2,1,4,3] => [2,1,4,3] => 1 = 3 - 2
[2,3,1,4] => [1,3,2,4] => 1 = 3 - 2
[2,3,4,1] => [1,2,4,3] => 1 = 3 - 2
[2,4,1,3] => [2,4,1,3] => 0 = 2 - 2
[2,4,3,1] => [1,4,3,2] => 1 = 3 - 2
[3,1,2,4] => [3,1,2,4] => 0 = 2 - 2
[3,1,4,2] => [2,1,4,3] => 1 = 3 - 2
[3,2,1,4] => [3,2,1,4] => 0 = 2 - 2
[3,2,4,1] => [2,1,4,3] => 1 = 3 - 2
[3,4,1,2] => [2,4,1,3] => 0 = 2 - 2
[3,4,2,1] => [1,4,3,2] => 1 = 3 - 2
[4,1,2,3] => [4,1,2,3] => 0 = 2 - 2
[4,1,3,2] => [4,1,3,2] => 1 = 3 - 2
[4,2,1,3] => [4,2,1,3] => 0 = 2 - 2
[4,2,3,1] => [4,1,3,2] => 1 = 3 - 2
[4,3,1,2] => [4,3,1,2] => 0 = 2 - 2
[4,3,2,1] => [4,3,2,1] => 0 = 2 - 2
Description
The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St001712
(load all 11 compositions to match this statistic)
Mapping
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
 => 0 = 2 - 2
[1,2] => [[1,2]]
 => 0 = 2 - 2
[2,1] => [[1],[2]]
 => 0 = 2 - 2
[1,2,3] => [[1,2,3]]
 => 0 = 2 - 2
[1,3,2] => [[1,2],[3]]
 => 0 = 2 - 2
[2,1,3] => [[1,3],[2]]
 => 1 = 3 - 2
[2,3,1] => [[1,2],[3]]
 => 0 = 2 - 2
[3,1,2] => [[1,3],[2]]
 => 1 = 3 - 2
[3,2,1] => [[1],[2],[3]]
 => 0 = 2 - 2
[1,2,3,4] => [[1,2,3,4]]
 => 0 = 2 - 2
[1,2,4,3] => [[1,2,3],[4]]
 => 0 = 2 - 2
[1,3,2,4] => [[1,2,4],[3]]
 => 1 = 3 - 2
[1,3,4,2] => [[1,2,3],[4]]
 => 0 = 2 - 2
[1,4,2,3] => [[1,2,4],[3]]
 => 1 = 3 - 2
[1,4,3,2] => [[1,2],[3],[4]]
 => 0 = 2 - 2
[2,1,3,4] => [[1,3,4],[2]]
 => 1 = 3 - 2
[2,1,4,3] => [[1,3],[2,4]]
 => 1 = 3 - 2
[2,3,1,4] => [[1,2,4],[3]]
 => 1 = 3 - 2
[2,3,4,1] => [[1,2,3],[4]]
 => 0 = 2 - 2
[2,4,1,3] => [[1,2],[3,4]]
 => 0 = 2 - 2
[2,4,3,1] => [[1,2],[3],[4]]
 => 0 = 2 - 2
[3,1,2,4] => [[1,3,4],[2]]
 => 1 = 3 - 2
[3,1,4,2] => [[1,3],[2,4]]
 => 1 = 3 - 2
[3,2,1,4] => [[1,4],[2],[3]]
 => 1 = 3 - 2
[3,2,4,1] => [[1,3],[2],[4]]
 => 1 = 3 - 2
[3,4,1,2] => [[1,2],[3,4]]
 => 0 = 2 - 2
[3,4,2,1] => [[1,2],[3],[4]]
 => 0 = 2 - 2
[4,1,2,3] => [[1,3,4],[2]]
 => 1 = 3 - 2
[4,1,3,2] => [[1,3],[2],[4]]
 => 1 = 3 - 2
[4,2,1,3] => [[1,4],[2],[3]]
 => 1 = 3 - 2
[4,2,3,1] => [[1,3],[2],[4]]
 => 1 = 3 - 2
[4,3,1,2] => [[1,4],[2],[3]]
 => 1 = 3 - 2
[4,3,2,1] => [[1],[2],[3],[4]]
 => 0 = 2 - 2
Description
The number of natural descents of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St001801
(load all 34 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [1,2] => 0 = 2 - 2
[2,1] => [1,2] => 0 = 2 - 2
[1,2,3] => [1,2,3] => 0 = 2 - 2
[1,3,2] => [1,2,3] => 0 = 2 - 2
[2,1,3] => [1,2,3] => 0 = 2 - 2
[2,3,1] => [1,2,3] => 0 = 2 - 2
[3,1,2] => [1,3,2] => 1 = 3 - 2
[3,2,1] => [1,3,2] => 1 = 3 - 2
[1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,2,4,3] => [1,2,3,4] => 0 = 2 - 2
[1,3,2,4] => [1,2,3,4] => 0 = 2 - 2
[1,3,4,2] => [1,2,3,4] => 0 = 2 - 2
[1,4,2,3] => [1,2,4,3] => 1 = 3 - 2
[1,4,3,2] => [1,2,4,3] => 1 = 3 - 2
[2,1,3,4] => [1,2,3,4] => 0 = 2 - 2
[2,1,4,3] => [1,2,3,4] => 0 = 2 - 2
[2,3,1,4] => [1,2,3,4] => 0 = 2 - 2
[2,3,4,1] => [1,2,3,4] => 0 = 2 - 2
[2,4,1,3] => [1,2,4,3] => 1 = 3 - 2
[2,4,3,1] => [1,2,4,3] => 1 = 3 - 2
[3,1,2,4] => [1,3,2,4] => 1 = 3 - 2
[3,1,4,2] => [1,3,4,2] => 1 = 3 - 2
[3,2,1,4] => [1,3,2,4] => 1 = 3 - 2
[3,2,4,1] => [1,3,4,2] => 1 = 3 - 2
[3,4,1,2] => [1,3,2,4] => 1 = 3 - 2
[3,4,2,1] => [1,3,2,4] => 1 = 3 - 2
[4,1,2,3] => [1,4,3,2] => 0 = 2 - 2
[4,1,3,2] => [1,4,2,3] => 1 = 3 - 2
[4,2,1,3] => [1,4,3,2] => 0 = 2 - 2
[4,2,3,1] => [1,4,2,3] => 1 = 3 - 2
[4,3,1,2] => [1,4,2,3] => 1 = 3 - 2
[4,3,2,1] => [1,4,2,3] => 1 = 3 - 2
Description
Half the number of preimage-image pairs of different parity in a permutation.
Matching statistic: St000318
(load all 17 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => 2
[1,2] => [1,2] => [2]
 => 2
[2,1] => [2,1] => [1,1]
 => 2
[1,2,3] => [1,2,3] => [3]
 => 2
[1,3,2] => [1,3,2] => [2,1]
 => 3
[2,1,3] => [2,1,3] => [2,1]
 => 3
[2,3,1] => [3,2,1] => [1,1,1]
 => 2
[3,1,2] => [3,2,1] => [1,1,1]
 => 2
[3,2,1] => [3,2,1] => [1,1,1]
 => 2
[1,2,3,4] => [1,2,3,4] => [4]
 => 2
[1,2,4,3] => [1,2,4,3] => [3,1]
 => 3
[1,3,2,4] => [1,3,2,4] => [3,1]
 => 3
[1,3,4,2] => [1,4,3,2] => [2,1,1]
 => 3
[1,4,2,3] => [1,4,3,2] => [2,1,1]
 => 3
[1,4,3,2] => [1,4,3,2] => [2,1,1]
 => 3
[2,1,3,4] => [2,1,3,4] => [3,1]
 => 3
[2,1,4,3] => [2,1,4,3] => [2,2]
 => 2
[2,3,1,4] => [3,2,1,4] => [2,1,1]
 => 3
[2,3,4,1] => [4,2,3,1] => [2,1,1]
 => 3
[2,4,1,3] => [3,4,1,2] => [2,2]
 => 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1]
 => 2
[3,1,2,4] => [3,2,1,4] => [2,1,1]
 => 3
[3,1,4,2] => [4,2,3,1] => [2,1,1]
 => 3
[3,2,1,4] => [3,2,1,4] => [2,1,1]
 => 3
[3,2,4,1] => [4,2,3,1] => [2,1,1]
 => 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1]
 => 2
[3,4,2,1] => [4,3,2,1] => [1,1,1,1]
 => 2
[4,1,2,3] => [4,2,3,1] => [2,1,1]
 => 3
[4,1,3,2] => [4,2,3,1] => [2,1,1]
 => 3
[4,2,1,3] => [4,3,2,1] => [1,1,1,1]
 => 2
[4,2,3,1] => [4,3,2,1] => [1,1,1,1]
 => 2
[4,3,1,2] => [4,3,2,1] => [1,1,1,1]
 => 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000759
(load all 2 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => 2
[1,2] => [1,2] => [1,1]
 => 2
[2,1] => [1,2] => [1,1]
 => 2
[1,2,3] => [1,2,3] => [1,1,1]
 => 2
[1,3,2] => [1,2,3] => [1,1,1]
 => 2
[2,1,3] => [1,2,3] => [1,1,1]
 => 2
[2,3,1] => [1,2,3] => [1,1,1]
 => 2
[3,1,2] => [1,3,2] => [2,1]
 => 3
[3,2,1] => [1,3,2] => [2,1]
 => 3
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 2
[1,2,4,3] => [1,2,3,4] => [1,1,1,1]
 => 2
[1,3,2,4] => [1,2,3,4] => [1,1,1,1]
 => 2
[1,3,4,2] => [1,2,3,4] => [1,1,1,1]
 => 2
[1,4,2,3] => [1,2,4,3] => [2,1,1]
 => 3
[1,4,3,2] => [1,2,4,3] => [2,1,1]
 => 3
[2,1,3,4] => [1,2,3,4] => [1,1,1,1]
 => 2
[2,1,4,3] => [1,2,3,4] => [1,1,1,1]
 => 2
[2,3,1,4] => [1,2,3,4] => [1,1,1,1]
 => 2
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 2
[2,4,1,3] => [1,2,4,3] => [2,1,1]
 => 3
[2,4,3,1] => [1,2,4,3] => [2,1,1]
 => 3
[3,1,2,4] => [1,3,2,4] => [2,1,1]
 => 3
[3,1,4,2] => [1,3,4,2] => [2,1,1]
 => 3
[3,2,1,4] => [1,3,2,4] => [2,1,1]
 => 3
[3,2,4,1] => [1,3,4,2] => [2,1,1]
 => 3
[3,4,1,2] => [1,3,2,4] => [2,1,1]
 => 3
[3,4,2,1] => [1,3,2,4] => [2,1,1]
 => 3
[4,1,2,3] => [1,4,3,2] => [3,1]
 => 2
[4,1,3,2] => [1,4,2,3] => [2,1,1]
 => 3
[4,2,1,3] => [1,4,3,2] => [3,1]
 => 2
[4,2,3,1] => [1,4,2,3] => [2,1,1]
 => 3
[4,3,1,2] => [1,4,2,3] => [2,1,1]
 => 3
[4,3,2,1] => [1,4,2,3] => [2,1,1]
 => 3
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000007
(load all 2 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 2 - 1
[1,2] => [1,2] => [1,2] => 1 = 2 - 1
[2,1] => [1,2] => [1,2] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [2,3,1] => 2 = 3 - 1
[2,1,3] => [1,3,2] => [2,3,1] => 2 = 3 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[3,1,2] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 2 = 3 - 1
[1,3,4,2] => [1,3,4,2] => [2,3,1,4] => 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 2 = 3 - 1
[1,4,3,2] => [1,4,2,3] => [3,4,1,2] => 2 = 3 - 1
[2,1,3,4] => [1,3,4,2] => [2,3,1,4] => 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => [3,4,1,2] => 2 = 3 - 1
[2,3,1,4] => [1,4,2,3] => [3,4,1,2] => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[2,4,1,3] => [1,3,2,4] => [1,2,4,3] => 2 = 3 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 2 = 3 - 1
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => 2 = 3 - 1
[3,1,4,2] => [1,4,2,3] => [3,4,1,2] => 2 = 3 - 1
[3,2,1,4] => [1,4,2,3] => [3,4,1,2] => 2 = 3 - 1
[3,2,4,1] => [1,2,4,3] => [2,3,4,1] => 2 = 3 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[4,1,3,2] => [1,3,2,4] => [1,2,4,3] => 2 = 3 - 1
[4,2,1,3] => [1,3,2,4] => [1,2,4,3] => 2 = 3 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
The following 566 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000031The number of cycles in the cycle decomposition of a permutation. St000099The number of valleys of a permutation, including the boundary. St000159The number of distinct parts of the integer partition. St000201The number of leaf nodes in a binary tree. St000325The width of the tree associated to a permutation. St000388The number of orbits of vertices of a graph under automorphisms. St000396The register function (or Horton-Strahler number) of a binary tree. St000470The number of runs in a permutation. St000522The number of 1-protected nodes of a rooted tree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000862The number of parts of the shifted shape of a permutation. St000903The number of different parts of an integer composition. St000920The logarithmic height of a Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001484The number of singletons of an integer partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001716The 1-improper chromatic number of a graph. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000039The number of crossings of a permutation. St000142The number of even parts of a partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000237The number of small exceedances. St000317The cycle descent number of a permutation. St000356The number of occurrences of the pattern 13-2. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000731The number of double exceedences of a permutation. St000834The number of right outer peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. 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. 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. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001071The beta invariant of the graph. St001083The number of boxed occurrences of 132 in a permutation. St001092The number of distinct even parts of a partition. 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. St001252Half the sum of the even parts of a partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001394The genus of a permutation. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001565The number of arithmetic progressions of length 2 in a permutation. 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. St001657The number of twos in an integer partition. St001665The number of pure excedances of a permutation. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of 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. St000015The number of peaks of a Dyck path. St000397The Strahler number of a rooted tree. St000630The length of the shortest palindromic decomposition of a binary word. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000891The number of distinct diagonal sums of a permutation matrix. 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 $c_{n-1}$. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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 $eA$. 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: 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000010The length of the partition. St000053The number of valleys of the Dyck path. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000105The number of blocks in the set partition. St000153The number of adjacent cycles of a permutation. St000183The side length of the Durfee square of an integer partition. St000216The absolute length of a permutation. St000239The number of small weak excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000298The order dimension or Dushnik-Miller dimension of a poset. St000306The bounce count of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. 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. St000659The number of rises of length at least 2 of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000679The pruning number of an ordered tree. St000703The number of deficiencies of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000876The number of factors in the Catalan decomposition of a binary word. St000889The number of alternating sign matrices with the same antidiagonal sums. St000905The number of different multiplicities of parts of an integer composition. St000917The open packing number of a graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001057The Grundy value of the game of creating an independent set in a graph. 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. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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. St001151The number of blocks with odd minimum. 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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 $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. 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 $eA$. 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. 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. St001235The global dimension of the corresponding Comp-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. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001367The smallest number which does not occur as degree of a vertex in a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001432The order dimension of the partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001597The Frobenius rank of a skew partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St001732The number of peaks visible from the left. St001734The lettericity of a graph. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001828The Euler characteristic of a graph. 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). St001884The number of borders of a binary word. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000017The number of inversions of a standard tableau. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000052The number of valleys of a Dyck path not on the x-axis. St000091The descent variation of a composition. St000117The number of centered tunnels of a Dyck path. St000143The largest repeated part 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000245The number of ascents of a permutation. 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. St000274The number of perfect matchings of a graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000441The number of successions of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000523The number of 2-protected nodes of a rooted tree. St000534The number of 2-rises of a permutation. St000547The number of even non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000628The balance of a binary word. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000665The number of rafts of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000691The number of changes of a binary word. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St000769The major index of a composition regarded as a word. St000779The tier of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000877The depth of the binary word interpreted as a path. St000992The alternating sum of the parts of an integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001175The size of a partition minus the hook length of the base cell. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001214The aft of an integer partition. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001395The number of strictly unfriendly partitions of a graph. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001459The number of zero columns in the nullspace of a graph. St001470The cyclic holeyness of a permutation. St001479The number of bridges of a graph. St001513The number of nested exceedences of a permutation. St001584The area statistic between a Dyck path and its bounce path. St001673The degree of asymmetry of an integer composition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 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. St001726The number of visible inversions of a permutation. St001731The factorization defect of a permutation. St001743The discrepancy of a graph. St001777The number of weak descents in an integer composition. 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. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001895The oddness of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001931The weak major index of an integer composition regarded as a word. St001960The number of descents of a permutation minus one if its first entry is not one. St000486The number of cycles of length at least 3 of a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St000711The number of big exceedences of a permutation. St001424The number of distinct squares in a binary word. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000886The number of permutations with the same antidiagonal sums. St000988The orbit size of a permutation under Foata's bijection. St000354The number of recoils of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000836The number of descents of distance 2 of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000504The cardinality of the first block of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000640The rank of the largest boolean interval in a poset. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000702The number of weak deficiencies of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001052The length of the exterior of a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001642The Prague dimension of a graph. St000295The length of the border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000461The rix statistic of a permutation. St000491The number of inversions of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. 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. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) 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. St000658The number of rises of length 2 of a Dyck path. St000661The number of rises of length 3 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. St000732The number of double deficiencies of a permutation. St000753The Grundy value for the game of Kayles on a binary word. 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. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. 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. 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. St001141The number of occurrences of hills of size 3 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001280The number of parts of an integer partition that are at least two. St001298The number of repeated entries in the Lehmer code of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001811The Castelnuovo-Mumford regularity of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001948The number of augmented double ascents of a permutation. St001330The hat guessing number of a graph. St001820The size of the image of the pop stack sorting operator. St001624The breadth of a lattice. St000219The number of occurrences of the pattern 231 in a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. 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 $eA$. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St000307The number of rowmotion orbits of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001964The interval resolution global dimension of a poset. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000944The 3-degree of an integer partition. St000456The monochromatic index of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001118The acyclic chromatic index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000264The girth of a graph, which is not a tree. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000422The energy of a graph, if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected 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. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000842The breadth of a permutation. St001875The number of simple modules with projective dimension at most 1. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001316The domatic number of a graph. 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. St000916The packing number of a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001272The number of graphs with the same degree sequence. St001393The induced matching number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a 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. 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. St000315The number of isolated vertices of a graph. St000906The length of the shortest maximal chain in a poset. St001352The number of internal nodes in the modular decomposition of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. 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. St001481The minimal height of a peak of a Dyck path. St001518The number of graphs with the same ordinary spectrum as the given graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St001949The rigidity index of a graph. 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. St000252The number of nodes of degree 3 of a binary tree. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. 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 $4$. St000310The minimal degree of a vertex 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. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. 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. St001573The minimal number of edges to remove to make a graph triangle-free. 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. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001488The number of corners of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer 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. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000102The charge of a semistandard tableau. St000822The Hadwiger number of the graph. 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. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. 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. St001572The minimal number of edges to remove to make a graph bipartite. 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. St001783The number of odd automorphisms of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001856The number of edges in the reduced word graph of a permutation.