Your data matches 286 different statistics following compositions of up to 3 maps.
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Matching statistic: St000017
(load all 85 compositions to match this statistic)
Mapping
St000017: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => 0 = 1 - 1
[[1,2]]
 => 0 = 1 - 1
[[1],[2]]
 => 0 = 1 - 1
[[1,2,3]]
 => 0 = 1 - 1
[[1,3],[2]]
 => 0 = 1 - 1
[[1,2],[3]]
 => 0 = 1 - 1
[[1],[2],[3]]
 => 0 = 1 - 1
[[1,2,3,4]]
 => 0 = 1 - 1
[[1,3,4],[2]]
 => 0 = 1 - 1
[[1,2,4],[3]]
 => 0 = 1 - 1
[[1,2,3],[4]]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => 1 = 2 - 1
[[1,2],[3,4]]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => 0 = 1 - 1
[[1,2],[3],[4]]
 => 0 = 1 - 1
[[1],[2],[3],[4]]
 => 0 = 1 - 1
[[1,2,3,4,5]]
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => 0 = 1 - 1
[[1,2,4,5],[3]]
 => 0 = 1 - 1
[[1,2,3,5],[4]]
 => 0 = 1 - 1
[[1,2,3,4],[5]]
 => 0 = 1 - 1
[[1,3,5],[2,4]]
 => 1 = 2 - 1
[[1,2,5],[3,4]]
 => 1 = 2 - 1
[[1,3,4],[2,5]]
 => 1 = 2 - 1
[[1,2,4],[3,5]]
 => 1 = 2 - 1
[[1,2,3],[4,5]]
 => 1 = 2 - 1
[[1,4,5],[2],[3]]
 => 0 = 1 - 1
[[1,3,5],[2],[4]]
 => 0 = 1 - 1
[[1,2,5],[3],[4]]
 => 0 = 1 - 1
[[1,3,4],[2],[5]]
 => 0 = 1 - 1
[[1,2,4],[3],[5]]
 => 0 = 1 - 1
[[1,2,3],[4],[5]]
 => 0 = 1 - 1
[[1,4],[2,5],[3]]
 => 1 = 2 - 1
[[1,3],[2,5],[4]]
 => 1 = 2 - 1
[[1,2],[3,5],[4]]
 => 1 = 2 - 1
[[1,3],[2,4],[5]]
 => 1 = 2 - 1
[[1,2],[3,4],[5]]
 => 1 = 2 - 1
[[1,5],[2],[3],[4]]
 => 0 = 1 - 1
[[1,4],[2],[3],[5]]
 => 0 = 1 - 1
[[1,3],[2],[4],[5]]
 => 0 = 1 - 1
[[1,2],[3],[4],[5]]
 => 0 = 1 - 1
[[1],[2],[3],[4],[5]]
 => 0 = 1 - 1
Description
The number of inversions of a standard tableau. Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.
Matching statistic: St000099
(load all 41 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => 1
[[1,2]]
 => [1,2] => 1
[[1],[2]]
 => [2,1] => 1
[[1,2,3]]
 => [1,2,3] => 1
[[1,3],[2]]
 => [2,1,3] => 1
[[1,2],[3]]
 => [3,1,2] => 1
[[1],[2],[3]]
 => [3,2,1] => 1
[[1,2,3,4]]
 => [1,2,3,4] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => 1
[[1,2,4],[3]]
 => [3,1,2,4] => 1
[[1,2,3],[4]]
 => [4,1,2,3] => 1
[[1,3],[2,4]]
 => [2,4,1,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => 1
[[1,2],[3],[4]]
 => [4,3,1,2] => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => 2
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 2
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1
Description
The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000183
(load all 3 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => 1
[[1,2]]
 => [2]
 => 1
[[1],[2]]
 => [1,1]
 => 1
[[1,2,3]]
 => [3]
 => 1
[[1,3],[2]]
 => [2,1]
 => 1
[[1,2],[3]]
 => [2,1]
 => 1
[[1],[2],[3]]
 => [1,1,1]
 => 1
[[1,2,3,4]]
 => [4]
 => 1
[[1,3,4],[2]]
 => [3,1]
 => 1
[[1,2,4],[3]]
 => [3,1]
 => 1
[[1,2,3],[4]]
 => [3,1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => 2
[[1,2],[3,4]]
 => [2,2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => 1
[[1,3],[2],[4]]
 => [2,1,1]
 => 1
[[1,2],[3],[4]]
 => [2,1,1]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => 1
[[1,2,3,4,5]]
 => [5]
 => 1
[[1,3,4,5],[2]]
 => [4,1]
 => 1
[[1,2,4,5],[3]]
 => [4,1]
 => 1
[[1,2,3,5],[4]]
 => [4,1]
 => 1
[[1,2,3,4],[5]]
 => [4,1]
 => 1
[[1,3,5],[2,4]]
 => [3,2]
 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => 1
[[1,3,5],[2],[4]]
 => [3,1,1]
 => 1
[[1,2,5],[3],[4]]
 => [3,1,1]
 => 1
[[1,3,4],[2],[5]]
 => [3,1,1]
 => 1
[[1,2,4],[3],[5]]
 => [3,1,1]
 => 1
[[1,2,3],[4],[5]]
 => [3,1,1]
 => 1
[[1,4],[2,5],[3]]
 => [2,2,1]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => 2
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => 1
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => 1
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => 1
Description
The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank.
Matching statistic: St000023
(load all 41 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => 0 = 1 - 1
[[1,2]]
 => [1,2] => 0 = 1 - 1
[[1],[2]]
 => [2,1] => 0 = 1 - 1
[[1,2,3]]
 => [1,2,3] => 0 = 1 - 1
[[1,3],[2]]
 => [2,1,3] => 0 = 1 - 1
[[1,2],[3]]
 => [3,1,2] => 0 = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => 0 = 1 - 1
[[1,2,3,4]]
 => [1,2,3,4] => 0 = 1 - 1
[[1,3,4],[2]]
 => [2,1,3,4] => 0 = 1 - 1
[[1,2,4],[3]]
 => [3,1,2,4] => 0 = 1 - 1
[[1,2,3],[4]]
 => [4,1,2,3] => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,4,1,3] => 1 = 2 - 1
[[1,2],[3,4]]
 => [3,4,1,2] => 1 = 2 - 1
[[1,4],[2],[3]]
 => [3,2,1,4] => 0 = 1 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3],[4]]
 => [4,3,1,2] => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => 0 = 1 - 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => 0 = 1 - 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => 0 = 1 - 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => 0 = 1 - 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => 0 = 1 - 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => 1 = 2 - 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => 1 = 2 - 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => 1 = 2 - 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => 1 = 2 - 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1 = 2 - 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 0 = 1 - 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => 0 = 1 - 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 0 = 1 - 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => 0 = 1 - 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 0 = 1 - 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 0 = 1 - 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => 1 = 2 - 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 1 = 2 - 1
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => 1 = 2 - 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => 1 = 2 - 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1 = 2 - 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 0 = 1 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => 0 = 1 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => 0 = 1 - 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 0 = 1 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 0 = 1 - 1
Description
The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].
Matching statistic: St001175
(load all 3 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => 0 = 1 - 1
[[1,2]]
 => [2]
 => 0 = 1 - 1
[[1],[2]]
 => [1,1]
 => 0 = 1 - 1
[[1,2,3]]
 => [3]
 => 0 = 1 - 1
[[1,3],[2]]
 => [2,1]
 => 0 = 1 - 1
[[1,2],[3]]
 => [2,1]
 => 0 = 1 - 1
[[1],[2],[3]]
 => [1,1,1]
 => 0 = 1 - 1
[[1,2,3,4]]
 => [4]
 => 0 = 1 - 1
[[1,3,4],[2]]
 => [3,1]
 => 0 = 1 - 1
[[1,2,4],[3]]
 => [3,1]
 => 0 = 1 - 1
[[1,2,3],[4]]
 => [3,1]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,2]
 => 1 = 2 - 1
[[1,2],[3,4]]
 => [2,2]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => 0 = 1 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => 0 = 1 - 1
[[1,2,3,4,5]]
 => [5]
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => [4,1]
 => 0 = 1 - 1
[[1,2,4,5],[3]]
 => [4,1]
 => 0 = 1 - 1
[[1,2,3,5],[4]]
 => [4,1]
 => 0 = 1 - 1
[[1,2,3,4],[5]]
 => [4,1]
 => 0 = 1 - 1
[[1,3,5],[2,4]]
 => [3,2]
 => 1 = 2 - 1
[[1,2,5],[3,4]]
 => [3,2]
 => 1 = 2 - 1
[[1,3,4],[2,5]]
 => [3,2]
 => 1 = 2 - 1
[[1,2,4],[3,5]]
 => [3,2]
 => 1 = 2 - 1
[[1,2,3],[4,5]]
 => [3,2]
 => 1 = 2 - 1
[[1,4,5],[2],[3]]
 => [3,1,1]
 => 0 = 1 - 1
[[1,3,5],[2],[4]]
 => [3,1,1]
 => 0 = 1 - 1
[[1,2,5],[3],[4]]
 => [3,1,1]
 => 0 = 1 - 1
[[1,3,4],[2],[5]]
 => [3,1,1]
 => 0 = 1 - 1
[[1,2,4],[3],[5]]
 => [3,1,1]
 => 0 = 1 - 1
[[1,2,3],[4],[5]]
 => [3,1,1]
 => 0 = 1 - 1
[[1,4],[2,5],[3]]
 => [2,2,1]
 => 1 = 2 - 1
[[1,3],[2,5],[4]]
 => [2,2,1]
 => 1 = 2 - 1
[[1,2],[3,5],[4]]
 => [2,2,1]
 => 1 = 2 - 1
[[1,3],[2,4],[5]]
 => [2,2,1]
 => 1 = 2 - 1
[[1,2],[3,4],[5]]
 => [2,2,1]
 => 1 = 2 - 1
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => 0 = 1 - 1
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => 0 = 1 - 1
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => 0 = 1 - 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => 0 = 1 - 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => 0 = 1 - 1
Description
The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000092
(load all 17 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => 1
[[1,2]]
 => [1,2] => [1,2] => 1
[[1],[2]]
 => [2,1] => [2,1] => 1
[[1,2,3]]
 => [1,2,3] => [1,2,3] => 1
[[1,3],[2]]
 => [2,1,3] => [2,3,1] => 1
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,3,4,1] => 1
[[1,2,4],[3]]
 => [3,1,2,4] => [1,3,4,2] => 1
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => 1
[[1,3],[2,4]]
 => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,4,2,1] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,3,1] => 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,3,4,5,1] => 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,3,4,5,2] => 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,4,5,3] => 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,4,5,1,3] => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,5,1,2] => 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,3,5,1,4] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,3,5,2,4] => 2
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,4,5,2,3] => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,4,5,2,1] => 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [2,4,5,3,1] => 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,4,5,3,2] => 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,3,5,4,1] => 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,3,5,4,2] => 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [3,5,2,4,1] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,5,2,3,1] => 2
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [4,5,1,3,2] => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [2,5,1,4,3] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [3,5,1,4,2] => 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,5,3,2,1] => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [3,5,4,2,1] => 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [2,5,4,3,1] => 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,5,4,3,2] => 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1
Description
The number of outer peaks of a permutation. An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$. In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000321
(load all 2 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => []
 => 1
[[1,2]]
 => [2]
 => []
 => 1
[[1],[2]]
 => [1,1]
 => [1]
 => 1
[[1,2,3]]
 => [3]
 => []
 => 1
[[1,3],[2]]
 => [2,1]
 => [1]
 => 1
[[1,2],[3]]
 => [2,1]
 => [1]
 => 1
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3,4]]
 => [4]
 => []
 => 1
[[1,3,4],[2]]
 => [3,1]
 => [1]
 => 1
[[1,2,4],[3]]
 => [3,1]
 => [1]
 => 1
[[1,2,3],[4]]
 => [3,1]
 => [1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => 2
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1,2,3,4,5]]
 => [5]
 => []
 => 1
[[1,3,4,5],[2]]
 => [4,1]
 => [1]
 => 1
[[1,2,4,5],[3]]
 => [4,1]
 => [1]
 => 1
[[1,2,3,5],[4]]
 => [4,1]
 => [1]
 => 1
[[1,2,3,4],[5]]
 => [4,1]
 => [1]
 => 1
[[1,3,5],[2,4]]
 => [3,2]
 => [2]
 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => [2]
 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => [2]
 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => [2]
 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => [2]
 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
Description
The number of integer partitions of n that are dominated by an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.
Matching statistic: St000345
(load all 2 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => []
 => 1
[[1,2]]
 => [2]
 => []
 => 1
[[1],[2]]
 => [1,1]
 => [1]
 => 1
[[1,2,3]]
 => [3]
 => []
 => 1
[[1,3],[2]]
 => [2,1]
 => [1]
 => 1
[[1,2],[3]]
 => [2,1]
 => [1]
 => 1
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3,4]]
 => [4]
 => []
 => 1
[[1,3,4],[2]]
 => [3,1]
 => [1]
 => 1
[[1,2,4],[3]]
 => [3,1]
 => [1]
 => 1
[[1,2,3],[4]]
 => [3,1]
 => [1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => 2
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1,2,3,4,5]]
 => [5]
 => []
 => 1
[[1,3,4,5],[2]]
 => [4,1]
 => [1]
 => 1
[[1,2,4,5],[3]]
 => [4,1]
 => [1]
 => 1
[[1,2,3,5],[4]]
 => [4,1]
 => [1]
 => 1
[[1,2,3,4],[5]]
 => [4,1]
 => [1]
 => 1
[[1,3,5],[2,4]]
 => [3,2]
 => [2]
 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => [2]
 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => [2]
 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => [2]
 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => [2]
 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,1]
 => 1
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [2,1]
 => 2
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
Description
The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000862
(load all 25 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => 1
[[1,2]]
 => [1,2] => [1,2] => 1
[[1],[2]]
 => [2,1] => [2,1] => 1
[[1,2,3]]
 => [1,2,3] => [1,2,3] => 1
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => 1
[[1,2],[3]]
 => [3,1,2] => [2,3,1] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 1
[[1,2,4],[3]]
 => [3,1,2,4] => [2,3,1,4] => 1
[[1,2,3],[4]]
 => [4,1,2,3] => [2,3,4,1] => 1
[[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,2,4,1] => 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,4,2,1] => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,3,1,4,5] => 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,3,4,1,5] => 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [2,3,4,5,1] => 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [3,1,4,2,5] => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,1,2,5] => 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [3,1,4,5,2] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [3,4,1,5,2] => 2
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,4,5,1,2] => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [3,2,4,1,5] => 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [3,4,2,1,5] => 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [3,2,4,5,1] => 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [3,4,2,5,1] => 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,4,5,2,1] => 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [4,2,1,5,3] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,2,5,1,3] => 2
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [4,5,2,1,3] => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [4,2,5,3,1] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [4,5,2,3,1] => 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,3,2,5,1] => 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [4,3,5,2,1] => 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [4,5,3,2,1] => 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1
Description
The number of parts of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the number of parts of the shifted shape.
Matching statistic: St000897
(load all 4 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => [1]
 => 1
[[1,2]]
 => [2]
 => [1,1]
 => 1
[[1],[2]]
 => [1,1]
 => [2]
 => 1
[[1,2,3]]
 => [3]
 => [2,1]
 => 1
[[1,3],[2]]
 => [2,1]
 => [2,1]
 => 1
[[1,2],[3]]
 => [2,1]
 => [2,1]
 => 1
[[1],[2],[3]]
 => [1,1,1]
 => [3]
 => 1
[[1,2,3,4]]
 => [4]
 => [3,1]
 => 1
[[1,3,4],[2]]
 => [3,1]
 => [2,2]
 => 1
[[1,2,4],[3]]
 => [3,1]
 => [2,2]
 => 1
[[1,2,3],[4]]
 => [3,1]
 => [2,2]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2,1,1]
 => 2
[[1,2],[3,4]]
 => [2,2]
 => [2,1,1]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [3,1]
 => 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [3,1]
 => 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [3,1]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [4]
 => 1
[[1,2,3,4,5]]
 => [5]
 => [4,1]
 => 1
[[1,3,4,5],[2]]
 => [4,1]
 => [3,2]
 => 1
[[1,2,4,5],[3]]
 => [4,1]
 => [3,2]
 => 1
[[1,2,3,5],[4]]
 => [4,1]
 => [3,2]
 => 1
[[1,2,3,4],[5]]
 => [4,1]
 => [3,2]
 => 1
[[1,3,5],[2,4]]
 => [3,2]
 => [2,2,1]
 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => [2,2,1]
 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => [2,2,1]
 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => [2,2,1]
 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => [2,2,1]
 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [3,2]
 => 1
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [3,2]
 => 1
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [3,2]
 => 1
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [3,2]
 => 1
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [3,2]
 => 1
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [3,2]
 => 1
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [3,1,1]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [3,1,1]
 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [3,1,1]
 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [3,1,1]
 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [3,1,1]
 => 2
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [4,1]
 => 1
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [4,1]
 => 1
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [4,1]
 => 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [4,1]
 => 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [5]
 => 1
Description
The number of different multiplicities of parts of an integer partition.
The following 276 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001597The Frobenius rank of a skew partition. St000057The Shynar inversion number of a standard tableau. St000142The number of even parts of a partition. St000204The number of internal nodes of a binary tree. 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. St000660The number of rises of length at least 3 of a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001092The number of distinct even parts of a partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001596The number of two-by-two squares inside a skew partition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001840The number of descents of a set partition. St000001The number of reduced words for a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000201The number of leaf nodes in a binary tree. St000291The number of descents of a binary word. St000331The number of upper interactions of a Dyck path. St000346The number of coarsenings of a partition. St000352The Elizalde-Pak rank of a permutation. St000390The number of runs of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000679The pruning number of an ordered tree. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000767The number of runs in an integer composition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000816The number of standard composition tableaux of the composition. St000820The number of compositions obtained by rotating the composition. St000829The Ulam distance of a permutation to the identity permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000903The number of different parts of an integer composition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. 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 $K[x]/(x^n)$. St001220The width of a permutation. 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. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001282The number of graphs with the same chromatic polynomial. St001313The number of Dyck paths above the lattice path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001427The number of descents of a signed permutation. 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. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001735The number of permutations with the same set of runs. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001874Lusztig's a-function for the symmetric group. 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. St000002The number of occurrences of the pattern 123 in a permutation. St000009The charge of a standard tableau. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000217The number of occurrences of the pattern 312 in a permutation. St000223The number of nestings in the permutation. St000252The number of nodes of degree 3 of a binary tree. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000481The number of upper covers of a partition in dominance order. St000534The number of 2-rises of a permutation. St000552The number of cut vertices of a graph. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000682The Grundy value of Welter's game on a binary word. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000879The number of long braid edges in the graph of braid moves of a permutation. 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. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 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. 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. St001172The number of 1-rises at odd height of a Dyck path. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001394The genus of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001673The degree of asymmetry of an integer composition. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001712The number of natural descents of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001777The number of weak descents in an integer composition. St001839The number of excedances of a set partition. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001931The weak major index of an integer composition regarded as a word. 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. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000568The hook number of a binary tree. St000886The number of permutations with the same antidiagonal sums. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000661The number of rises of length 3 of a Dyck path. St000779The tier of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000640The rank of the largest boolean interval in a poset. St000847The number of standard Young tableaux whose descent set is the binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000369The dinv deficit of a Dyck path. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000516The number of stretching pairs of a permutation. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) 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. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. 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. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001856The number of edges in the reduced word graph of a permutation. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. 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. St000326The position of the first one in a binary word after appending a 1 at the end. St000378The diagonal inversion number of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000734The last entry in the first row of a standard tableau. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001462The number of factors of a standard tableaux under concatenation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000016The number of attacking pairs of a standard tableau. 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. 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. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000877The depth of the binary word interpreted as a path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. 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 $eA$. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the 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. 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. St000944The 3-degree of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. 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. 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. St001868The number of alignments of type NE of a signed permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001866The nesting alignments of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000264The girth of a graph, which is not a tree. St001964The interval resolution global dimension of a poset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001890The maximum magnitude of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001060The distinguishing index of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St001857The number of edges in the reduced word graph of a signed permutation. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000478Another weight of a partition according to Alladi. St000936The number of even values of the symmetric group character corresponding to the partition. St001926Sparre Andersen's position of the maximum of a signed permutation. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching.