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Your data matches 221 different statistics following compositions of up to 3 maps.
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Matching statistic: St000384
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1 = 2 - 1
[1,2] => [2]
=> 2 = 3 - 1
[2,1] => [1,1]
=> 2 = 3 - 1
[1,2,3] => [3]
=> 3 = 4 - 1
[1,3,2] => [2,1]
=> 2 = 3 - 1
[2,1,3] => [2,1]
=> 2 = 3 - 1
[2,3,1] => [2,1]
=> 2 = 3 - 1
[3,1,2] => [2,1]
=> 2 = 3 - 1
[3,2,1] => [1,1,1]
=> 3 = 4 - 1
[1,2,4,3] => [3,1]
=> 3 = 4 - 1
[1,3,2,4] => [3,1]
=> 3 = 4 - 1
[1,3,4,2] => [3,1]
=> 3 = 4 - 1
[1,4,2,3] => [3,1]
=> 3 = 4 - 1
[1,4,3,2] => [2,1,1]
=> 3 = 4 - 1
[2,1,3,4] => [3,1]
=> 3 = 4 - 1
[2,1,4,3] => [2,2]
=> 3 = 4 - 1
[2,3,1,4] => [3,1]
=> 3 = 4 - 1
[2,3,4,1] => [3,1]
=> 3 = 4 - 1
[2,4,1,3] => [2,2]
=> 3 = 4 - 1
[2,4,3,1] => [2,1,1]
=> 3 = 4 - 1
[3,1,2,4] => [3,1]
=> 3 = 4 - 1
[3,1,4,2] => [2,2]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> 3 = 4 - 1
[3,2,4,1] => [2,1,1]
=> 3 = 4 - 1
[3,4,1,2] => [2,2]
=> 3 = 4 - 1
[3,4,2,1] => [2,1,1]
=> 3 = 4 - 1
[4,1,2,3] => [3,1]
=> 3 = 4 - 1
[4,1,3,2] => [2,1,1]
=> 3 = 4 - 1
[4,2,1,3] => [2,1,1]
=> 3 = 4 - 1
[4,2,3,1] => [2,1,1]
=> 3 = 4 - 1
[4,3,1,2] => [2,1,1]
=> 3 = 4 - 1
[1,2,5,4,3] => [3,1,1]
=> 3 = 4 - 1
[1,3,2,5,4] => [3,2]
=> 3 = 4 - 1
[1,3,5,2,4] => [3,2]
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1]
=> 3 = 4 - 1
[1,4,2,5,3] => [3,2]
=> 3 = 4 - 1
[1,4,3,2,5] => [3,1,1]
=> 3 = 4 - 1
[1,4,3,5,2] => [3,1,1]
=> 3 = 4 - 1
[1,4,5,2,3] => [3,2]
=> 3 = 4 - 1
[1,4,5,3,2] => [3,1,1]
=> 3 = 4 - 1
[1,5,2,4,3] => [3,1,1]
=> 3 = 4 - 1
[1,5,3,2,4] => [3,1,1]
=> 3 = 4 - 1
[1,5,3,4,2] => [3,1,1]
=> 3 = 4 - 1
[1,5,4,2,3] => [3,1,1]
=> 3 = 4 - 1
[2,1,3,5,4] => [3,2]
=> 3 = 4 - 1
[2,1,4,3,5] => [3,2]
=> 3 = 4 - 1
[2,1,4,5,3] => [3,2]
=> 3 = 4 - 1
[2,1,5,3,4] => [3,2]
=> 3 = 4 - 1
[2,1,5,4,3] => [2,2,1]
=> 3 = 4 - 1
[2,3,1,5,4] => [3,2]
=> 3 = 4 - 1
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St001515
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,2] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St001020
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> 2
[1,2] => [2]
=> [1,1,0,0,1,0]
=> 3
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 3
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
[] => []
=> []
=> ? = 1
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001190
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001190: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001190: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> 3 = 2 + 1
[1,2] => [2]
=> [1,1,0,0,1,0]
=> 4 = 3 + 1
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[] => []
=> []
=> ? = 1 + 1
Description
Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra.
Matching statistic: St001650
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001650: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001650: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> 3 = 2 + 1
[1,2] => [2]
=> [1,1,0,0,1,0]
=> 4 = 3 + 1
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[] => []
=> []
=> ? = 1 + 1
Description
The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000144
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,2] => [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[] => []
=> []
=> []
=> ? = 1
Description
The pyramid weight of the Dyck path.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St000216
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> [3,1,2] => 2
[1,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 4
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 4
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 4
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[] => []
=> []
=> [1] => ? = 1
Description
The absolute length of a permutation.
The absolute length of a permutation $\sigma$ of length $n$ is the shortest $k$ such that $\sigma = t_1 \dots t_k$ for transpositions $t_i$. Also, this is equal to $n$ minus the number of cycles of $\sigma$.
Matching statistic: St000288
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> 1010 => 2
[1,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 3
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 3
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 4
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 3
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 4
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 4
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 4
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 4
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 4
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 4
[] => []
=> []
=> => ? = 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000336
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[1,2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 4
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 4
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 4
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 4
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 4
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 4
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 4
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 4
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 4
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 4
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 4
[] => []
=> []
=> ?
=> ? = 1
Description
The leg major index of a standard tableau.
The leg length of a cell is the number of cells strictly below in the same column. This statistic is the sum of all leg lengths. Therefore, this is actually a statistic on the underlying integer partition.
It happens to coincide with the (leg) major index of a tabloid restricted to standard Young tableaux, defined as follows: the descent set of a tabloid is the set of cells, not in the top row, whose entry is strictly larger than the entry directly above it. The leg major index is the sum of the leg lengths of the descents plus the number of descents.
Matching statistic: St000395
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,2] => [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[] => []
=> []
=> []
=> ? = 1
Description
The sum of the heights of the peaks of a Dyck path.
The following 211 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000501The size of the first part in the decomposition of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000809The reduced reflection length of the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000924The number of topologically connected components of a perfect matching. St000957The number of Bruhat lower covers of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001480The number of simple summands of the module J^2/J^3. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000019The cardinality of the support of a permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000050The depth or height of a binary tree. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000528The height of a poset. St000673The number of non-fixed points of a permutation. St000863The length of the first row of the shifted shape of a permutation. 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. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001958The degree of the polynomial interpolating the values of a permutation. St000080The rank of the poset. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001344The neighbouring number of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000962The 3-shifted major index of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000735The last entry on the main diagonal of a standard tableau. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer 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. St001901The largest multiplicity of an irreducible representation 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer 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. 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. 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. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent 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. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000259The diameter of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000706The product of the factorials of the multiplicities of an integer 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. 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. St001568The smallest positive integer that does not appear twice in the partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000934The 2-degree of an integer partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001875The number of simple modules with projective dimension at most 1. 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. St001877Number of indecomposable injective modules with projective dimension 2. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000699The toughness times the least common multiple of 1,. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000782The indicator function of whether a given perfect matching is an L & P matching.
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