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Your data matches 800 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> 1
[2,1]
=> 1
[1,1,1]
=> 2
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 2
[4,1]
=> 1
[3,2]
=> 2
[3,1,1]
=> 2
[2,2,1]
=> 3
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 2
[3,2,1]
=> 3
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 2
[4,2,1]
=> 3
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 2
[5,2,1]
=> 3
[8,1]
=> 1
[7,2]
=> 2
[7,1,1]
=> 2
[6,2,1]
=> 3
[9,1]
=> 1
[8,2]
=> 2
[8,1,1]
=> 2
[7,2,1]
=> 3
[10,1]
=> 1
[9,2]
=> 2
[9,1,1]
=> 2
[8,2,1]
=> 3
[11,1]
=> 1
[10,2]
=> 2
[10,1,1]
=> 2
[9,2,1]
=> 3
[12,1]
=> 1
[11,2]
=> 2
[11,1,1]
=> 2
[10,2,1]
=> 3
[13,1]
=> 1
[12,2]
=> 2
[12,1,1]
=> 2
[11,2,1]
=> 3
[14,1]
=> 1
[13,2]
=> 2
[13,1,1]
=> 2
[12,2,1]
=> 3
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000179
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000179: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000179: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> 1
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 2
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 2
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 2
[2,2,1]
=> [2,1]
=> 3
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> 3
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> 3
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> 3
[8,1]
=> [1]
=> 1
[7,2]
=> [2]
=> 2
[7,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> 3
[9,1]
=> [1]
=> 1
[8,2]
=> [2]
=> 2
[8,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> 3
[10,1]
=> [1]
=> 1
[9,2]
=> [2]
=> 2
[9,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> 3
[11,1]
=> [1]
=> 1
[10,2]
=> [2]
=> 2
[10,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> 3
[12,1]
=> [1]
=> 1
[11,2]
=> [2]
=> 2
[11,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> 3
[13,1]
=> [1]
=> 1
[12,2]
=> [2]
=> 2
[12,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> 3
[14,1]
=> [1]
=> 1
[13,2]
=> [2]
=> 2
[13,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> 3
Description
The product of the hook lengths of the integer partition.
Consider the Ferrers diagram associated with the integer partition. For each cell in the diagram, drawn using the English convention, consider its ''hook'': the cell itself, all cells in the same row to the right and all cells in the same column below. The ''hook length of a cell'' is the number of cells in the hook of a cell. This statistic is the product of the hook lengths of all cells in the partition.
Let $H_\lambda$ denote this product, then the number of standard Young tableaux of shape $\lambda$, (traditionally denoted $f^\lambda$) equals $n! / H_\lambda$. Therefore, it is consistent to set the product of the hook lengths of the empty partition equal to $1$.
Matching statistic: St000228
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> 1
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 2
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 2
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 2
[2,2,1]
=> [2,1]
=> 3
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> 3
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> 3
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> 3
[8,1]
=> [1]
=> 1
[7,2]
=> [2]
=> 2
[7,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> 3
[9,1]
=> [1]
=> 1
[8,2]
=> [2]
=> 2
[8,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> 3
[10,1]
=> [1]
=> 1
[9,2]
=> [2]
=> 2
[9,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> 3
[11,1]
=> [1]
=> 1
[10,2]
=> [2]
=> 2
[10,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> 3
[12,1]
=> [1]
=> 1
[11,2]
=> [2]
=> 2
[11,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> 3
[13,1]
=> [1]
=> 1
[12,2]
=> [2]
=> 2
[12,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> 3
[14,1]
=> [1]
=> 1
[13,2]
=> [2]
=> 2
[13,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000459
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> 1
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 2
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 2
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 2
[2,2,1]
=> [2,1]
=> 3
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> 3
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> 3
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> 3
[8,1]
=> [1]
=> 1
[7,2]
=> [2]
=> 2
[7,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> 3
[9,1]
=> [1]
=> 1
[8,2]
=> [2]
=> 2
[8,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> 3
[10,1]
=> [1]
=> 1
[9,2]
=> [2]
=> 2
[9,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> 3
[11,1]
=> [1]
=> 1
[10,2]
=> [2]
=> 2
[10,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> 3
[12,1]
=> [1]
=> 1
[11,2]
=> [2]
=> 2
[11,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> 3
[13,1]
=> [1]
=> 1
[12,2]
=> [2]
=> 2
[12,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> 3
[14,1]
=> [1]
=> 1
[13,2]
=> [2]
=> 2
[13,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> 3
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> 1
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 2
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 2
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 2
[2,2,1]
=> [2,1]
=> 3
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> 3
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> 3
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> 3
[8,1]
=> [1]
=> 1
[7,2]
=> [2]
=> 2
[7,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> 3
[9,1]
=> [1]
=> 1
[8,2]
=> [2]
=> 2
[8,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> 3
[10,1]
=> [1]
=> 1
[9,2]
=> [2]
=> 2
[9,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> 3
[11,1]
=> [1]
=> 1
[10,2]
=> [2]
=> 2
[10,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> 3
[12,1]
=> [1]
=> 1
[11,2]
=> [2]
=> 2
[11,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> 3
[13,1]
=> [1]
=> 1
[12,2]
=> [2]
=> 2
[12,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> 3
[14,1]
=> [1]
=> 1
[13,2]
=> [2]
=> 2
[13,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> 3
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> 1
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 2
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 2
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 2
[2,2,1]
=> [2,1]
=> 3
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> 3
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> 3
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> 3
[8,1]
=> [1]
=> 1
[7,2]
=> [2]
=> 2
[7,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> 3
[9,1]
=> [1]
=> 1
[8,2]
=> [2]
=> 2
[8,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> 3
[10,1]
=> [1]
=> 1
[9,2]
=> [2]
=> 2
[9,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> 3
[11,1]
=> [1]
=> 1
[10,2]
=> [2]
=> 2
[10,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> 3
[12,1]
=> [1]
=> 1
[11,2]
=> [2]
=> 2
[11,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> 3
[13,1]
=> [1]
=> 1
[12,2]
=> [2]
=> 2
[12,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> 3
[14,1]
=> [1]
=> 1
[13,2]
=> [2]
=> 2
[13,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> 3
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001380
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> 1
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 2
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 2
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 2
[2,2,1]
=> [2,1]
=> 3
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> 3
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> 3
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> 3
[8,1]
=> [1]
=> 1
[7,2]
=> [2]
=> 2
[7,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> 3
[9,1]
=> [1]
=> 1
[8,2]
=> [2]
=> 2
[8,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> 3
[10,1]
=> [1]
=> 1
[9,2]
=> [2]
=> 2
[9,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> 3
[11,1]
=> [1]
=> 1
[10,2]
=> [2]
=> 2
[10,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> 3
[12,1]
=> [1]
=> 1
[11,2]
=> [2]
=> 2
[11,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> 3
[13,1]
=> [1]
=> 1
[12,2]
=> [2]
=> 2
[12,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> 3
[14,1]
=> [1]
=> 1
[13,2]
=> [2]
=> 2
[13,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> 3
Description
The number of monomer-dimer tilings of a Ferrers diagram.
For a hook of length $n$, this is the $n$-th Fibonacci number.
Matching statistic: St001382
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001382: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> 0 = 1 - 1
[2,1]
=> [1]
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> 1 = 2 - 1
[3,1]
=> [1]
=> 0 = 1 - 1
[2,2]
=> [2]
=> 1 = 2 - 1
[2,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,1]
=> [1]
=> 0 = 1 - 1
[3,2]
=> [2]
=> 1 = 2 - 1
[3,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,2,1]
=> [2,1]
=> 2 = 3 - 1
[5,1]
=> [1]
=> 0 = 1 - 1
[4,2]
=> [2]
=> 1 = 2 - 1
[4,1,1]
=> [1,1]
=> 1 = 2 - 1
[3,2,1]
=> [2,1]
=> 2 = 3 - 1
[6,1]
=> [1]
=> 0 = 1 - 1
[5,2]
=> [2]
=> 1 = 2 - 1
[5,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,2,1]
=> [2,1]
=> 2 = 3 - 1
[7,1]
=> [1]
=> 0 = 1 - 1
[6,2]
=> [2]
=> 1 = 2 - 1
[6,1,1]
=> [1,1]
=> 1 = 2 - 1
[5,2,1]
=> [2,1]
=> 2 = 3 - 1
[8,1]
=> [1]
=> 0 = 1 - 1
[7,2]
=> [2]
=> 1 = 2 - 1
[7,1,1]
=> [1,1]
=> 1 = 2 - 1
[6,2,1]
=> [2,1]
=> 2 = 3 - 1
[9,1]
=> [1]
=> 0 = 1 - 1
[8,2]
=> [2]
=> 1 = 2 - 1
[8,1,1]
=> [1,1]
=> 1 = 2 - 1
[7,2,1]
=> [2,1]
=> 2 = 3 - 1
[10,1]
=> [1]
=> 0 = 1 - 1
[9,2]
=> [2]
=> 1 = 2 - 1
[9,1,1]
=> [1,1]
=> 1 = 2 - 1
[8,2,1]
=> [2,1]
=> 2 = 3 - 1
[11,1]
=> [1]
=> 0 = 1 - 1
[10,2]
=> [2]
=> 1 = 2 - 1
[10,1,1]
=> [1,1]
=> 1 = 2 - 1
[9,2,1]
=> [2,1]
=> 2 = 3 - 1
[12,1]
=> [1]
=> 0 = 1 - 1
[11,2]
=> [2]
=> 1 = 2 - 1
[11,1,1]
=> [1,1]
=> 1 = 2 - 1
[10,2,1]
=> [2,1]
=> 2 = 3 - 1
[13,1]
=> [1]
=> 0 = 1 - 1
[12,2]
=> [2]
=> 1 = 2 - 1
[12,1,1]
=> [1,1]
=> 1 = 2 - 1
[11,2,1]
=> [2,1]
=> 2 = 3 - 1
[14,1]
=> [1]
=> 0 = 1 - 1
[13,2]
=> [2]
=> 1 = 2 - 1
[13,1,1]
=> [1,1]
=> 1 = 2 - 1
[12,2,1]
=> [2,1]
=> 2 = 3 - 1
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Matching statistic: St000144
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0]
=> 1
[2,1]
=> [1]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,1]
=> [1]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,1]
=> [1]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[5,1]
=> [1]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[6,1]
=> [1]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[7,1]
=> [1]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[8,1]
=> [1]
=> [1,0]
=> 1
[7,2]
=> [2]
=> [1,0,1,0]
=> 2
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[9,1]
=> [1]
=> [1,0]
=> 1
[8,2]
=> [2]
=> [1,0,1,0]
=> 2
[8,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[10,1]
=> [1]
=> [1,0]
=> 1
[9,2]
=> [2]
=> [1,0,1,0]
=> 2
[9,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[11,1]
=> [1]
=> [1,0]
=> 1
[10,2]
=> [2]
=> [1,0,1,0]
=> 2
[10,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[12,1]
=> [1]
=> [1,0]
=> 1
[11,2]
=> [2]
=> [1,0,1,0]
=> 2
[11,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[10,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[13,1]
=> [1]
=> [1,0]
=> 1
[12,2]
=> [2]
=> [1,0,1,0]
=> 2
[12,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[11,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[14,1]
=> [1]
=> [1,0]
=> 1
[13,2]
=> [2]
=> [1,0,1,0]
=> 2
[13,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[12,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
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: St000184
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000184: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000184: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1]
=> 1
[2,1]
=> [1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,1]
=> [1]
=> [1]
=> 1
[2,2]
=> [2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> [1,1]
=> 2
[4,1]
=> [1]
=> [1]
=> 1
[3,2]
=> [2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> [1,1]
=> 2
[2,2,1]
=> [2,1]
=> [3]
=> 3
[5,1]
=> [1]
=> [1]
=> 1
[4,2]
=> [2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [3]
=> 3
[6,1]
=> [1]
=> [1]
=> 1
[5,2]
=> [2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [3]
=> 3
[7,1]
=> [1]
=> [1]
=> 1
[6,2]
=> [2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [3]
=> 3
[8,1]
=> [1]
=> [1]
=> 1
[7,2]
=> [2]
=> [2]
=> 2
[7,1,1]
=> [1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> [3]
=> 3
[9,1]
=> [1]
=> [1]
=> 1
[8,2]
=> [2]
=> [2]
=> 2
[8,1,1]
=> [1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> [3]
=> 3
[10,1]
=> [1]
=> [1]
=> 1
[9,2]
=> [2]
=> [2]
=> 2
[9,1,1]
=> [1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> [3]
=> 3
[11,1]
=> [1]
=> [1]
=> 1
[10,2]
=> [2]
=> [2]
=> 2
[10,1,1]
=> [1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> [3]
=> 3
[12,1]
=> [1]
=> [1]
=> 1
[11,2]
=> [2]
=> [2]
=> 2
[11,1,1]
=> [1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> [3]
=> 3
[13,1]
=> [1]
=> [1]
=> 1
[12,2]
=> [2]
=> [2]
=> 2
[12,1,1]
=> [1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> [3]
=> 3
[14,1]
=> [1]
=> [1]
=> 1
[13,2]
=> [2]
=> [2]
=> 2
[13,1,1]
=> [1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> [3]
=> 3
Description
The size of the centralizer of any permutation of given cycle type.
The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$:
$$C_g = \{h \in G : hgh^{-1} = g\}.$$
Its size thus depends only on the conjugacy class of $g$.
The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is
$$|C| = \Pi j^{a_j} a_j!$$
For example, for any permutation with cycle type $\lambda = (3,2,2,1)$,
$$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$
There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$.
The following 790 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000293The number of inversions of a binary word. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000631The number of distinct palindromic decompositions of a binary word. St000784The maximum of the length and the largest part of the integer partition. St000922The minimal number such that all substrings of this length are unique. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001360The number of covering relations in Young's lattice below a partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001488The number of corners of a skew partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000921The number of internal inversions of a binary word. 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. St001012Number of simple modules with projective dimension at most 2 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001400The total number of Littlewood-Richardson tableaux of given shape. St001437The flex of a binary word. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. 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. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000013The height of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000050The depth or height of a binary tree. St000075The orbit size of a standard tableau under promotion. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000189The number of elements in the poset. St000197The number of entries equal to positive one in the alternating sign matrix. St000203The number of external nodes of a binary tree. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000236The number of cyclical small weak excedances. St000246The number of non-inversions of a permutation. St000288The number of ones in a binary word. St000290The major index of a binary word. St000335The difference of lower and upper interactions. St000336The leg major index of a standard tableau. St000369The dinv deficit of a Dyck path. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000458The number of permutations obtained by switching adjacencies or successions. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000507The number of ascents of a standard tableau. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000657The smallest part of an integer composition. St000670The reversal length of a permutation. St000690The size of the conjugacy class of a permutation. St000719The number of alignments in a perfect matching. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000734The last entry in the first row of a standard tableau. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744The length of the path to the largest entry in a standard Young tableau. St000780The size of the orbit under rotation of a perfect matching. St000808The number of up steps of the associated bargraph. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000863The length of the first row of the shifted shape of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000891The number of distinct diagonal sums of a permutation matrix. St000924The number of topologically connected components of a perfect matching. St000945The number of matchings in the dihedral orbit of a perfect matching. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000957The number of Bruhat lower covers of a permutation. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000982The length of the longest constant subword. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in 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. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001346The number of parking functions that give the same permutation. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001439The number of even weak deficiencies and of odd weak exceedences. St001462The number of factors of a standard tableaux under concatenation. St001480The number of simple summands of the module J^2/J^3. St001485The modular major index of a binary word. St001486The number of corners of the ribbon associated with an integer composition. St001500The global dimension of magnitude 1 Nakayama algebras. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001554The number of distinct nonempty subtrees of a binary tree. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001759The Rajchgot index of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001813The product of the sizes of the principal order filters in a poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001956The comajor index for set-valued two-row standard Young tableaux. St001958The degree of the polynomial interpolating the values of a permutation. St001959The product of the heights of the peaks of a Dyck path. St000009The charge of a standard tableau. St000012The area of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000026The position of the first return of a Dyck path. St000044The number of vertices of the unicellular map given by a perfect matching. St000058The order of a permutation. St000152The number of boxed plus the number of special entries. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000392The length of the longest run of ones in a binary word. St000398The sum of the depths of the vertices (or total internal path length) of a binary tree. St000420The number of Dyck paths that are weakly above a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000647The number of big descents of a permutation. St000651The maximal size of a rise in a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000673The number of non-fixed points of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000844The size of the largest block in the direct sum decomposition of a permutation. St000845The maximal number of elements covered by an element in a poset. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000981The length of the longest zigzag subpath. St000984The number of boxes below precisely one peak. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001160The number of proper blocks (or intervals) of a permutations. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001371The length of the longest Yamanouchi prefix of a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001658The total number of rook placements on a Ferrers board. St001721The degree of a binary word. St001925The minimal number of zeros in a row of an alternating sign matrix. St001930The weak major index of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. St000806The semiperimeter of the associated bargraph. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001468The smallest fixpoint of a permutation. St001838The number of nonempty primitive factors of a binary word. St000377The dinv defect of an integer partition. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000376The bounce deficit of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000411The tree factorial of a binary tree. St000625The sum of the minimal distances to a greater element. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000890The number of nonzero entries in an alternating sign matrix. St001074The number of inversions of the cyclic embedding of a permutation. St000060The greater neighbor of the maximum. St000100The number of linear extensions of a poset. St000327The number of cover relations in a poset. St000385The number of vertices with out-degree 1 in a binary tree. St000402Half the size of the symmetry class of a permutation. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000530The number of permutations with the same descent word as the given permutation. St000568The hook number of a binary tree. St000569The sum of the heights of the vertices of a binary tree. St000619The number of cyclic descents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000633The size of the automorphism group of a poset. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000886The number of permutations with the same antidiagonal sums. St000910The number of maximal chains of minimal length in a poset. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001313The number of Dyck paths above the lattice path given by a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St000219The number of occurrences of the pattern 231 in a permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000538The number of even inversions of a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. 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. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. 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. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. 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. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000747A variant of the major index of a set partition. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000931The number of occurrences of the pattern UUU in a Dyck path. St000961The shifted major index of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001141The number of occurrences of hills of size 3 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 $K[x]/(x^n)$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001388The number of non-attacking neighbors of a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001557The number of inversions of the second entry of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001731The factorization defect of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000935The number of ordered refinements of an integer partition. St000010The length of the partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000547The number of even non-empty partial sums of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000147The largest part of an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001777The number of weak descents in an integer composition. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000738The first entry in the last row of a standard tableau. St000141The maximum drop size of a permutation. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St000074The number of special entries. St000157The number of descents of a standard tableau. St000245The number of ascents of a permutation. St000441The number of successions of a permutation. St000883The number of longest increasing subsequences of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000295The length of the border of a binary word. St000352The Elizalde-Pak rank of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001115The number of even descents of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St000007The number of saliances of the permutation. 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 $eA$. St000214The number of adjacencies of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001378The product of the cohook lengths of the integer partition. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. 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. St000839The largest opener of a set partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001090The number of pop-stack-sorts needed to sort a permutation. St000648The number of 2-excedences of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000345The number of refinements of a partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000234The number of global ascents of a permutation. St000313The number of degree 2 vertices of a graph. St000356The number of occurrences of the pattern 13-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000439The position of the first down step of a Dyck path. St000463The number of admissible inversions of a permutation. St000703The number of deficiencies of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St001489The maximum of the number of descents and the number of inverse descents. St001726The number of visible inversions of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000354The number of recoils of a permutation. St000539The number of odd inversions of a permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000956The maximal displacement of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001910The height of the middle non-run of a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St000223The number of nestings in the permutation. St000534The number of 2-rises of a permutation. St000503The maximal difference between two elements in a common block. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000056The decomposition (or block) number of a permutation. St000171The degree of the graph. St000211The rank of the set partition. St000240The number of indices that are not small excedances. St000325The width of the tree associated to a permutation. St000413The number of ordered trees with the same underlying unordered tree. St000446The disorder of a permutation. St000502The number of successions of a set partitions. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000728The dimension of a set partition. St000833The comajor index of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000991The number of right-to-left minima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001120The length of a longest path in a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001461The number of topologically connected components of the chord diagram of a permutation. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000053The number of valleys of the Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000258The burning number of a graph. St000273The domination number of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000339The maf index of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000505The biggest entry in the block containing the 1. St000544The cop number of a graph. St000552The number of cut vertices of a graph. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000915The Ore degree of a graph. St000916The packing number of a graph. St000947The major index east count of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001062The maximal size of a block of a set partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001093The detour number of a graph. St001110The 3-dynamic chromatic number of a graph. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001308The number of induced paths on three vertices in a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001463The number of distinct columns in the nullspace of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number of a graph. St001727The number of invisible inversions of a permutation. St001829The common independence number of a graph. St000081The number of edges of a graph. St000702The number of weak deficiencies of a permutation. St000874The position of the last double rise in a Dyck path. St000911The number of maximal antichains of maximal size in a poset. St001340The cardinality of a minimal non-edge isolating set of a graph. St000731The number of double exceedences of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001268The size of the largest ordinal summand in the poset. St000078The number of alternating sign matrices whose left key is the permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000652The maximal difference between successive positions of a permutation. St000730The maximal arc length of a set partition. St000740The last entry of a permutation. St001277The degeneracy of a graph. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000004The major index of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000344The number of strongly connected outdegree sequences of a graph. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000653The last descent of a permutation. St000722The number of different neighbourhoods in a graph. St000794The mak of a permutation. St000971The smallest closer of a set partition. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001781The interlacing number of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001874Lusztig's a-function for the symmetric group. St001963The tree-depth of a graph. St000255The number of reduced Kogan faces with the permutation as type. St000933The number of multipartitions of sizes given by an integer partition. St000553The number of blocks of a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000939The number of characters of the symmetric group whose value on the partition is positive. St001497The position of the largest weak excedence of a permutation. St000067The inversion number of the alternating sign matrix. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000235The number of indices that are not cyclical small weak excedances. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000015The number of peaks of a Dyck path. St000051The size of the left subtree of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St000654The first descent of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001220The width of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001298The number of repeated entries in the Lehmer code of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001530The depth of a Dyck path. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001649The length of a longest trail in a graph. St001869The maximum cut size of a graph. St000005The bounce statistic of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000083The number of left oriented leafs of a binary tree except the first one. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000120The number of left tunnels of a Dyck path. St000222The number of alignments in the permutation. St000242The number of indices that are not cyclical small weak excedances. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000331The number of upper interactions of a Dyck path. St000338The number of pixed points of a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000663The number of right floats of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001118The acyclic chromatic index of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St001644The dimension of a graph. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. 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. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. 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. St001383The BG-rank of an 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. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer 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. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. 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. 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. St001610The number of coloured endofunctions 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. 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. 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. 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. 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. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. 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. St000474Dyson's crank of a partition. St000099The number of valleys of a permutation, including the boundary. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000023The number of inner peaks of a permutation. St000039The number of crossings of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000822The Hadwiger number of the graph. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. 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. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001728The number of invisible descents of a permutation. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000840The number of closers smaller than the largest opener in a perfect matching. St000045The number of linear extensions of a binary tree. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. 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. St001569The maximal modular displacement of a permutation. St000284The Plancherel distribution on integer partitions. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000707The product of the factorials of the parts. St000713The dimension of the irreducible representation of Sp(4) labelled by 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. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St000929The constant term of the character polynomial of an integer 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. 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. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000789The number of crossing-similar perfect matchings of a perfect matching. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001330The hat guessing number of a graph. St001555The order of a signed permutation. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000177The number of free tiles in the pattern. St000178Number of free entries. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000739The first entry in the last row of a semistandard tableau. St001684The reduced word complexity of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001596The number of two-by-two squares inside a skew partition. St001821The sorting index of a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000735The last entry on the main diagonal of a standard tableau. St001651The Frankl number of a lattice. St000264The girth of a graph, which is not a tree. St001768The number of reduced words of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001863The number of weak excedances of a signed permutation. St000422The energy of a graph, if it is integral. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000456The monochromatic index of a connected graph. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree.
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