Your data matches 111 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001159
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001159: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
Description
Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St001640
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [2,1] => [1,2] => 1
[2]
 => [1,0,1,0]
 => [3,1,2] => [1,3,2] => 0
[1,1]
 => [1,1,0,0]
 => [2,3,1] => [2,1,3] => 1
[3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [1,2,4,3] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [2,5,4,1,3] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [3,2,1,4] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,5,2,1,4] => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,1,2,5,3] => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,3,2,5,4] => 0
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St001948
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [2,1] => [1,2] => 1
[2]
 => [1,0,1,0]
 => [3,1,2] => [2,1,3] => 0
[1,1]
 => [1,1,0,0]
 => [2,3,1] => [1,3,2] => 1
[3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [3,2,1,4] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [2,1,3,4] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [4,3,2,1,5] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [3,5,2,1,4] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [1,4,3,2] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [3,2,4,1,5] => 0
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2,1,4,5] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [2,5,4,1,3] => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [3,1,4,5,2] => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [2,1,4,3,5] => 0
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => 1
Description
The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Matching statistic: St001200
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1 + 2
[2]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[1,1]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? = 1 + 2
[3]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 0 + 2
[2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 1 + 2
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? = 1 + 2
[4]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 0 + 2
[3,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[2,2]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 1 + 2
[3,2]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 1 + 2
[3,3]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 0 + 2
[2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001199
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1,0]
 => [2,1] => [1,1,0,0]
 => ? = 1
[2]
 => [1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => ? = 0
[1,1]
 => [1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 1
[3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => ? = 0
[2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? = 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001498
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1,0]
 => [2,1] => [1,1,0,0]
 => ? = 1 - 1
[2]
 => [1,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => ? = 0 - 1
[1,1]
 => [1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => ? = 0 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 1 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 0 = 1 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000929
(load all 2 compositions to match this statistic)
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => []
 => ? = 1
[2]
 => [2]
 => []
 => ? = 0
[1,1]
 => [1,1]
 => [1]
 => ? = 1
[3]
 => [2,1]
 => [1]
 => ? = 0
[2,1]
 => [3]
 => []
 => ? = 1
[1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[4]
 => [2,2]
 => [2]
 => 0
[3,1]
 => [2,1,1]
 => [1,1]
 => 1
[2,2]
 => [4]
 => []
 => ? = 1
[2,1,1]
 => [3,1]
 => [1]
 => ? = 0
[1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,2]
 => [4,1]
 => [1]
 => ? = 1
[2,2,1]
 => [5]
 => []
 => ? = 1
[3,3]
 => [3,2,1]
 => [2,1]
 => 0
[2,2,2]
 => [6]
 => []
 => ? = 1
Description
The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001568
(load all 2 compositions to match this statistic)
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => []
 => ? = 1 + 1
[2]
 => [2]
 => []
 => ? = 0 + 1
[1,1]
 => [1,1]
 => [1]
 => ? = 1 + 1
[3]
 => [2,1]
 => [1]
 => ? = 0 + 1
[2,1]
 => [3]
 => []
 => ? = 1 + 1
[1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[4]
 => [2,2]
 => [2]
 => 1 = 0 + 1
[3,1]
 => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,2]
 => [4]
 => []
 => ? = 1 + 1
[2,1,1]
 => [3,1]
 => [1]
 => ? = 0 + 1
[1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[3,2]
 => [4,1]
 => [1]
 => ? = 1 + 1
[2,2,1]
 => [5]
 => []
 => ? = 1 + 1
[3,3]
 => [3,2,1]
 => [2,1]
 => 1 = 0 + 1
[2,2,2]
 => [6]
 => []
 => ? = 1 + 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000207
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Values
[1]
 => []
 => ?
 => ?
 => ? = 1
[2]
 => []
 => ?
 => ?
 => ? = 0
[1,1]
 => [1]
 => [1]
 => []
 => ? = 1
[3]
 => []
 => ?
 => ?
 => ? = 0
[2,1]
 => [1]
 => [1]
 => []
 => ? = 1
[1,1,1]
 => [1,1]
 => [2]
 => []
 => ? = 1
[4]
 => []
 => ?
 => ?
 => ? = 0
[3,1]
 => [1]
 => [1]
 => []
 => ? = 1
[2,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[2,1,1]
 => [1,1]
 => [2]
 => []
 => ? = 0
[1,1,1,1]
 => [1,1,1]
 => [2,1]
 => [1]
 => 1
[3,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[2,2,1]
 => [2,1]
 => [1,1,1]
 => [1,1]
 => 1
[3,3]
 => [3]
 => [3]
 => []
 => ? = 0
[2,2,2]
 => [2,2]
 => [1,1,1,1]
 => [1,1,1]
 => 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Values
[1]
 => []
 => ?
 => ?
 => ? = 1
[2]
 => []
 => ?
 => ?
 => ? = 0
[1,1]
 => [1]
 => [1]
 => []
 => ? = 1
[3]
 => []
 => ?
 => ?
 => ? = 0
[2,1]
 => [1]
 => [1]
 => []
 => ? = 1
[1,1,1]
 => [1,1]
 => [2]
 => []
 => ? = 1
[4]
 => []
 => ?
 => ?
 => ? = 0
[3,1]
 => [1]
 => [1]
 => []
 => ? = 1
[2,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[2,1,1]
 => [1,1]
 => [2]
 => []
 => ? = 0
[1,1,1,1]
 => [1,1,1]
 => [2,1]
 => [1]
 => 1
[3,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[2,2,1]
 => [2,1]
 => [1,1,1]
 => [1,1]
 => 1
[3,3]
 => [3]
 => [3]
 => []
 => ? = 0
[2,2,2]
 => [2,2]
 => [1,1,1,1]
 => [1,1,1]
 => 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].
The following 101 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000618The number of self-evacuating tableaux of given shape. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St000934The 2-degree of an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. 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. 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. 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. 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. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts 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. St000933The number of multipartitions of sizes given by an integer partition. 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. 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. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000873The aix statistic of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001153The number of blocks with even minimum in a set partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. 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. St000145The Dyson rank of a partition. St000823The number of unsplittable factors of the set partition. St000990The first ascent of a permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000474Dyson's crank of a partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St001128The exponens consonantiae of a partition. 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. St000478Another weight of a partition according to Alladi. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001545The second Elser number of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000264The girth of a graph, which is not a tree.