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Your data matches 128 different statistics following compositions of up to 3 maps.
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Matching statistic: St001175
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(load all 4 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[1,2] => [1,2] => [1,1]
=> 0
[1,-2] => [1,-2] => [1]
=> 0
[2,1] => [2,1] => [2]
=> 0
[2,-1] => [-1,2] => [1]
=> 0
[-2,1] => [-2,-1] => [2]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 0
[1,3,-2] => [1,-2,3] => [1,1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> 0
[1,-3,-2] => [1,-2,-3] => [1]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> 0
[2,1,-3] => [2,1,-3] => [2]
=> 0
[2,-1,3] => [-1,2,-3] => [1]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> 0
[-2,1,-3] => [-2,-1,-3] => [2]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> 0
[2,3,-1] => [-1,2,3] => [1,1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> 0
[2,-3,-1] => [-1,2,-3] => [1]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> 0
[-2,-3,1] => [-2,-1,-3] => [2]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> 0
[3,1,-2] => [3,-2,1] => [2]
=> 0
[3,-1,2] => [-1,-2,3] => [1]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> 0
[-3,1,-2] => [-3,-2,-1] => [2]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> 0
[3,2,-1] => [-1,3,2] => [2]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> 0
[3,-2,-1] => [-1,-2,3] => [1]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> 0
[-3,2,-1] => [-1,-3,-2] => [2]
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> 0
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000183
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1 = 0 + 1
[1,2] => [1,2] => [1,1]
=> 1 = 0 + 1
[1,-2] => [1,-2] => [1]
=> 1 = 0 + 1
[2,1] => [2,1] => [2]
=> 1 = 0 + 1
[2,-1] => [-1,2] => [1]
=> 1 = 0 + 1
[-2,1] => [-2,-1] => [2]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,2,-3] => [1,2,-3] => [1,1]
=> 1 = 0 + 1
[1,-2,3] => [1,-2,-3] => [1]
=> 1 = 0 + 1
[1,-2,-3] => [1,-2,-3] => [1]
=> 1 = 0 + 1
[-1,2,3] => [-1,-2,3] => [1]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
=> 1 = 0 + 1
[1,3,-2] => [1,-2,3] => [1,1]
=> 1 = 0 + 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> 1 = 0 + 1
[1,-3,-2] => [1,-2,-3] => [1]
=> 1 = 0 + 1
[-1,3,2] => [-1,-2,3] => [1]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1]
=> 1 = 0 + 1
[2,1,-3] => [2,1,-3] => [2]
=> 1 = 0 + 1
[2,-1,3] => [-1,2,-3] => [1]
=> 1 = 0 + 1
[2,-1,-3] => [-1,2,-3] => [1]
=> 1 = 0 + 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> 1 = 0 + 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> 1 = 0 + 1
[2,3,1] => [3,2,1] => [2,1]
=> 1 = 0 + 1
[2,3,-1] => [-1,2,3] => [1,1]
=> 1 = 0 + 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> 1 = 0 + 1
[2,-3,-1] => [-1,2,-3] => [1]
=> 1 = 0 + 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> 1 = 0 + 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> 1 = 0 + 1
[3,1,2] => [3,2,1] => [2,1]
=> 1 = 0 + 1
[3,1,-2] => [3,-2,1] => [2]
=> 1 = 0 + 1
[3,-1,2] => [-1,-2,3] => [1]
=> 1 = 0 + 1
[3,-1,-2] => [-1,-2,3] => [1]
=> 1 = 0 + 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> 1 = 0 + 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
=> 1 = 0 + 1
[3,2,-1] => [-1,3,2] => [2]
=> 1 = 0 + 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> 1 = 0 + 1
[3,-2,-1] => [-1,-2,3] => [1]
=> 1 = 0 + 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> 1 = 0 + 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> 1 = 0 + 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 1 = 0 + 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> 1 = 0 + 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 1 = 0 + 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> 1 = 0 + 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> 1 = 0 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> 1 = 0 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> 1 = 0 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> 1 = 0 + 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000017
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [[1]]
=> 0
[1,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 0
[1,-2] => [1,-2] => [1]
=> [[1]]
=> 0
[2,1] => [2,1] => [2]
=> [[1,2]]
=> 0
[2,-1] => [-1,2] => [1]
=> [[1]]
=> 0
[-2,1] => [-2,-1] => [2]
=> [[1,2]]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> [[1],[2]]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> [[1]]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> [[1]]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 0
[1,3,-2] => [1,-2,3] => [1,1]
=> [[1],[2]]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> [[1,2],[3]]
=> 0
[1,-3,-2] => [1,-2,-3] => [1]
=> [[1]]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 0
[2,1,-3] => [2,1,-3] => [2]
=> [[1,2]]
=> 0
[2,-1,3] => [-1,2,-3] => [1]
=> [[1]]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> [[1]]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [[1,2],[3]]
=> 0
[-2,1,-3] => [-2,-1,-3] => [2]
=> [[1,2]]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [[1,2],[3]]
=> 0
[2,3,-1] => [-1,2,3] => [1,1]
=> [[1],[2]]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> [[1,2],[3]]
=> 0
[2,-3,-1] => [-1,2,-3] => [1]
=> [[1]]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [[1,2],[3]]
=> 0
[-2,-3,1] => [-2,-1,-3] => [2]
=> [[1,2]]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [[1,2],[3]]
=> 0
[3,1,-2] => [3,-2,1] => [2]
=> [[1,2]]
=> 0
[3,-1,2] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [[1,2],[3]]
=> 0
[-3,1,-2] => [-3,-2,-1] => [2]
=> [[1,2]]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [[1,2],[3]]
=> 0
[3,2,-1] => [-1,3,2] => [2]
=> [[1,2]]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [[1,2],[3]]
=> 0
[3,-2,-1] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [[1,2],[3]]
=> 0
[-3,2,-1] => [-1,-3,-2] => [2]
=> [[1,2]]
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> [[1,2]]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> [[1]]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> [[1]]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> [[1]]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [[1],[2]]
=> 0
Description
The number of inversions of a standard tableau.
Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.
Matching statistic: St000057
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [[1]]
=> 0
[1,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 0
[1,-2] => [1,-2] => [1]
=> [[1]]
=> 0
[2,1] => [2,1] => [2]
=> [[1,2]]
=> 0
[2,-1] => [-1,2] => [1]
=> [[1]]
=> 0
[-2,1] => [-2,-1] => [2]
=> [[1,2]]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> [[1],[2]]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> [[1]]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> [[1]]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [[1,3],[2]]
=> 0
[1,3,-2] => [1,-2,3] => [1,1]
=> [[1],[2]]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> [[1,3],[2]]
=> 0
[1,-3,-2] => [1,-2,-3] => [1]
=> [[1]]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [[1,3],[2]]
=> 0
[2,1,-3] => [2,1,-3] => [2]
=> [[1,2]]
=> 0
[2,-1,3] => [-1,2,-3] => [1]
=> [[1]]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> [[1]]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [[1,3],[2]]
=> 0
[-2,1,-3] => [-2,-1,-3] => [2]
=> [[1,2]]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [[1,3],[2]]
=> 0
[2,3,-1] => [-1,2,3] => [1,1]
=> [[1],[2]]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> [[1,3],[2]]
=> 0
[2,-3,-1] => [-1,2,-3] => [1]
=> [[1]]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [[1,3],[2]]
=> 0
[-2,-3,1] => [-2,-1,-3] => [2]
=> [[1,2]]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [[1,3],[2]]
=> 0
[3,1,-2] => [3,-2,1] => [2]
=> [[1,2]]
=> 0
[3,-1,2] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [[1,3],[2]]
=> 0
[-3,1,-2] => [-3,-2,-1] => [2]
=> [[1,2]]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [[1,3],[2]]
=> 0
[3,2,-1] => [-1,3,2] => [2]
=> [[1,2]]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [[1,3],[2]]
=> 0
[3,-2,-1] => [-1,-2,3] => [1]
=> [[1]]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [[1,3],[2]]
=> 0
[-3,2,-1] => [-1,-3,-2] => [2]
=> [[1,2]]
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> [[1,2]]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> [[1]]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> [[1]]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> [[1]]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [[1],[2]]
=> 0
Description
The Shynar inversion number of a standard tableau.
Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Matching statistic: St000142
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> []
=> 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 0
[1,-2] => [1,-2] => [1]
=> []
=> 0
[2,1] => [2,1] => [2]
=> []
=> 0
[2,-1] => [-1,2] => [1]
=> []
=> 0
[-2,1] => [-2,-1] => [2]
=> []
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> []
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> []
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> []
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 0
[1,-3,-2] => [1,-2,-3] => [1]
=> []
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> []
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-3] => [2,1,-3] => [2]
=> []
=> 0
[2,-1,3] => [-1,2,-3] => [1]
=> []
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> []
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,1,-2] => [3,-2,1] => [2]
=> []
=> 0
[3,-1,2] => [-1,-2,3] => [1]
=> []
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> []
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 0
Description
The number of even parts of a partition.
Matching statistic: St000473
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> []
=> 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 0
[1,-2] => [1,-2] => [1]
=> []
=> 0
[2,1] => [2,1] => [2]
=> []
=> 0
[2,-1] => [-1,2] => [1]
=> []
=> 0
[-2,1] => [-2,-1] => [2]
=> []
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> []
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> []
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> []
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 0
[1,-3,-2] => [1,-2,-3] => [1]
=> []
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> []
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-3] => [2,1,-3] => [2]
=> []
=> 0
[2,-1,3] => [-1,2,-3] => [1]
=> []
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> []
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,1,-2] => [3,-2,1] => [2]
=> []
=> 0
[3,-1,2] => [-1,-2,3] => [1]
=> []
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> []
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 0
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000480
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> []
=> 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 0
[1,-2] => [1,-2] => [1]
=> []
=> 0
[2,1] => [2,1] => [2]
=> []
=> 0
[2,-1] => [-1,2] => [1]
=> []
=> 0
[-2,1] => [-2,-1] => [2]
=> []
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> []
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> []
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> []
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 0
[1,-3,-2] => [1,-2,-3] => [1]
=> []
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> []
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-3] => [2,1,-3] => [2]
=> []
=> 0
[2,-1,3] => [-1,2,-3] => [1]
=> []
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> []
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,1,-2] => [3,-2,1] => [2]
=> []
=> 0
[3,-1,2] => [-1,-2,3] => [1]
=> []
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> []
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 0
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000660
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1,0]
=> 0
[1,2] => [1,2] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2] => [1,-2] => [1]
=> [1,0]
=> 0
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 0
[2,-1] => [-1,2] => [1]
=> [1,0]
=> 0
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> [1,0]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> [1,0]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,-3,-2] => [1,-2,-3] => [1]
=> [1,0]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 0
[2,-1,3] => [-1,2,-3] => [1]
=> [1,0]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> [1,0]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,-3,-1] => [-1,2,-3] => [1]
=> [1,0]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 0
[3,-1,2] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,-2,-1] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 0
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001033
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1,0]
=> 0
[1,2] => [1,2] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2] => [1,-2] => [1]
=> [1,0]
=> 0
[2,1] => [2,1] => [2]
=> [1,0,1,0]
=> 0
[2,-1] => [-1,2] => [1]
=> [1,0]
=> 0
[-2,1] => [-2,-1] => [2]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> [1,0]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> [1,0]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,-3,-2] => [1,-2,-3] => [1]
=> [1,0]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,-3] => [2,1,-3] => [2]
=> [1,0,1,0]
=> 0
[2,-1,3] => [-1,2,-3] => [1]
=> [1,0]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> [1,0]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,-3,-1] => [-1,2,-3] => [1]
=> [1,0]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,-2] => [3,-2,1] => [2]
=> [1,0,1,0]
=> 0
[3,-1,2] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,0,1,0]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,-1] => [-1,3,2] => [2]
=> [1,0,1,0]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,-2,-1] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,0,1,0]
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 0
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001092
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> []
=> 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 0
[1,-2] => [1,-2] => [1]
=> []
=> 0
[2,1] => [2,1] => [2]
=> []
=> 0
[2,-1] => [-1,2] => [1]
=> []
=> 0
[-2,1] => [-2,-1] => [2]
=> []
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> []
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> []
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> []
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 0
[1,-3,-2] => [1,-2,-3] => [1]
=> []
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> []
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-3] => [2,1,-3] => [2]
=> []
=> 0
[2,-1,3] => [-1,2,-3] => [1]
=> []
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> []
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,1,-2] => [3,-2,1] => [2]
=> []
=> 0
[3,-1,2] => [-1,-2,3] => [1]
=> []
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> []
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> []
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 0
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
The following 118 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001596The number of two-by-two squares inside a skew partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St001597The Frobenius rank of a skew partition. St001731The factorization defect of a permutation. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. 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. 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. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the 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. St000369The dinv deficit of a Dyck path. St000567The sum of the products of all pairs of parts. St000674The number of hills of a Dyck path. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001141The number of occurrences of hills of size 3 in a Dyck path. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001248Sum of the even parts of a partition. St001279The sum of the 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. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000442The maximal area to the right of an up step of a Dyck path. St000618The number of self-evacuating tableaux of given shape. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St000444The length of the maximal rise of a Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001498The normalised height of a Nakayama algebra with magnitude 1. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001857The number of edges in the reduced word graph of a signed permutation. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000782The indicator function of whether a given perfect matching is an L & P matching.
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