Your data matches 213 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000602
(load all 3 compositions to match this statistic)
Mapping
St000602: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => 0
{{1},{2}}
 => 0
{{1,2,3}}
 => 0
{{1,2},{3}}
 => 0
{{1,3},{2}}
 => 1
{{1},{2,3}}
 => 0
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 0
{{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => 2
{{1,3},{2,4}}
 => 2
{{1,3},{2},{4}}
 => 1
{{1,4},{2,3}}
 => 2
{{1},{2,3,4}}
 => 0
{{1},{2,3},{4}}
 => 0
{{1,4},{2},{3}}
 => 2
{{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => 0
{{1},{2},{3},{4}}
 => 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal.
Matching statistic: St001699
(load all 8 compositions to match this statistic)
Mapping
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [[1,2]]
 => 0
{{1},{2}}
 => [[1],[2]]
 => 0
{{1,2,3}}
 => [[1,2,3]]
 => 0
{{1,2},{3}}
 => [[1,2],[3]]
 => 1
{{1,3},{2}}
 => [[1,3],[2]]
 => 0
{{1},{2,3}}
 => [[1,3],[2]]
 => 0
{{1},{2},{3}}
 => [[1],[2],[3]]
 => 0
{{1,2,3,4}}
 => [[1,2,3,4]]
 => 0
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 2
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 1
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => 0
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 2
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 0
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => 2
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 1
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => 2
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => 0
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 1
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => 0
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => 0
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 0
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 0
Description
The major index of a standard tableau minus the weighted size of its shape.
Matching statistic: St000034
(load all 16 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,1,3] => 0
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 1
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,1,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,3,4,1] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,4,2,1] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [3,1,2,4] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Matching statistic: St000218
(load all 20 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,1,3] => 1
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,3,4] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,1,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,3,4,1] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,4,2,1] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [3,1,2,4] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000220
(load all 25 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St000462
(load all 5 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000462: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
Description
The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000463
(load all 5 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,2,1] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,3,1] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,3,2,1] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,3,2,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of admissible inversions of a permutation. Let $w = w_1,w_2,\dots,w_k$ be a word of length $k$ with distinct letters from $[n]$. An admissible inversion of $w$ is a pair $(w_i,w_j)$ such that $1\leq i < j\leq k$ and $w_i > w_j$ that satisfies either of the following conditions: $1 < i$ and $w_{i−1} < w_i$ or there is some $l$ such that $i < l < j$ and $w_i < w_l$.
Matching statistic: St000483
(load all 5 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000483: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,2,4] => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of times a permutation switches from increasing to decreasing or decreasing to increasing. This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]
Matching statistic: St000799
(load all 8 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000799: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => 0
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [3,2,4,1] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,3,1] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,3,4,1] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [4,3,1,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,4,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [3,1,4,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,1,2,4] => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,3,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,1,3,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of occurrences of the vincular pattern |213 in a permutation. This is the number of occurrences of the pattern $(2,1,3)$, such that the letter matched by $2$ is the first entry of the permutation.
Matching statistic: St000800
(load all 13 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => 0
{{1,2},{3}}
 => [2,1,3] => [3,2,1] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [2,1,3] => 0
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [3,4,1,2] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,4,2,1] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1,2,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,1,4] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,2,4,1] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,3,1,2] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,2,1] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [2,4,1,3] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,3,1] => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,1,3,4] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,1,4] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => 2
Description
The number of occurrences of the vincular pattern |231 in a permutation. This is the number of occurrences of the pattern $(2,3,1)$, such that the letter matched by $2$ is the first entry of the permutation.
The following 203 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000833The comajor index of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001403The number of vertical separators in a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St000047The number of standard immaculate tableaux of a given shape. St000255The number of reduced Kogan faces with the permutation as type. St000638The number of up-down runs of a permutation. St000988The orbit size of a permutation under Foata's bijection. St000004The major index of a permutation. St000008The major index of the composition. 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. St000057The Shynar inversion number of a standard tableau. St000143The largest repeated part of a partition. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000169The cocharge of a standard tableau. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000209Maximum difference of elements in cycles. St000216The absolute length of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000225Difference between largest and smallest parts in a partition. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000355The number of occurrences of the pattern 21-3. St000376The bounce deficit of a Dyck path. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000446The disorder of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000572The dimension exponent of a set partition. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000653The last descent of a permutation. St000670The reversal length of a permutation. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000794The mak of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001082The number of boxed occurrences of 123 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001377The major index minus the number of inversions of a permutation. St001388The number of non-attacking neighbors of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001485The modular major index of a binary word. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001569The maximal modular displacement of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001684The reduced word complexity of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001867The number of alignments of type EN of a signed permutation. St000058The order of a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000485The length of the longest cycle of a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000844The size of the largest block in the direct sum decomposition of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000886The number of permutations with the same antidiagonal sums. St000983The length of the longest alternating subword. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001128The exponens consonantiae of a partition. St001246The maximal difference between two consecutive entries of a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001313The number of Dyck paths above the lattice path given by a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001486The number of corners of the ribbon associated with an integer composition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000219The number of occurrences of the pattern 231 in a permutation. St000045The number of linear extensions of a binary tree. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001964The interval resolution global dimension of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000567The sum of the products of all pairs of parts. St000674The number of hills of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000455The second largest eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,. St001651The Frankl number of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. 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. St001875The number of simple modules with projective dimension at most 1. St001857The number of edges in the reduced word graph of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001623The number of doubly irreducible elements of a lattice. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. 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. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001060The distinguishing index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.