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Matching statistic: St000159
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
St000159: 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
St000159: 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]
=> 1
[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]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[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]
=> 1
[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]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[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]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-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]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[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]
=> 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000340
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000340: 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
St000340: 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]
=> 1
[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]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[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]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[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]
=> 1
[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]
=> 1
[-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]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> [1,0]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-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]
=> 1
[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]
=> 1
[-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]
=> 1
[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]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-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]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[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]
=> 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000533
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
St000533: 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
St000533: 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]
=> 1
[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]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[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]
=> 1
[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]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[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]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-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]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[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]
=> 1
Description
The minimum of the number of parts and the size of the first part of an integer partition.
This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000783
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
St000783: 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
St000783: 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]
=> 1
[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]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[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]
=> 1
[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]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[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]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-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]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[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]
=> 1
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St001194
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001194: 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
St001194: 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]
=> 1
[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]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[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]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[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]
=> 1
[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]
=> 1
[-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]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1,1,0,0]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> [1,0]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-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]
=> 1
[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]
=> 1
[-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]
=> 1
[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]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> [1,0]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-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]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[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]
=> 1
Description
The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module
Matching statistic: St000318
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
St000318: 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
St000318: 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]
=> 2 = 1 + 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]
=> [1,1]
=> 2 = 1 + 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 2 = 1 + 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]
=> 2 = 1 + 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 2 = 1 + 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> 1 = 0 + 1
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 2 = 1 + 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 2 = 1 + 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 2 = 1 + 1
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> 1 = 0 + 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 2 = 1 + 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> 1 = 0 + 1
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 2 = 1 + 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]
=> 2 = 1 + 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 2 = 1 + 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> 1 = 0 + 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 2 = 1 + 1
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> 1 = 0 + 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 2 = 1 + 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]
=> [1,1,1]
=> 2 = 1 + 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 2 = 1 + 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 2 = 1 + 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 2 = 1 + 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]
=> 2 = 1 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St001432
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
St001432: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[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]
=> 1
[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]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> ? = 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[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]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> ? = 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> ? = 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> ? = 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-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]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[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]
=> 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[1,3,-4,-2] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,4,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-3,-4,2] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[1,-3,-4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,3,4,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,3,-4,2] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,2,-3] => [1,4,-3,2] => [2,1]
=> [1]
=> 1
[1,4,-2,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,4,-2,-3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-4,2,3] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,2,-3] => [1,-4,-3,-2] => [2,1]
=> [1]
=> 1
[1,-4,-2,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,4,2,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,-4,2,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,-2] => [1,-2,4,3] => [2,1]
=> [1]
=> 1
[1,4,-3,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-4,-3,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,4,3,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001780
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
St001780: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[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]
=> 1
[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]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> ? = 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[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]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> ? = 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> ? = 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> ? = 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-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]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[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]
=> 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[1,3,-4,-2] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,4,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-3,-4,2] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[1,-3,-4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,3,4,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,3,-4,2] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,2,-3] => [1,4,-3,2] => [2,1]
=> [1]
=> 1
[1,4,-2,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,4,-2,-3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-4,2,3] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,2,-3] => [1,-4,-3,-2] => [2,1]
=> [1]
=> 1
[1,-4,-2,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,4,2,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,-4,2,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,-2] => [1,-2,4,3] => [2,1]
=> [1]
=> 1
[1,4,-3,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-4,-3,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,4,3,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
Description
The order of promotion on the set of standard tableaux of given shape.
Matching statistic: St001899
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
St001899: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001899: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[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]
=> 1
[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]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> ? = 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[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]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> ? = 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> ? = 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> ? = 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-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]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[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]
=> 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[1,3,-4,-2] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,4,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-3,-4,2] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[1,-3,-4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,3,4,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,3,-4,2] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,2,-3] => [1,4,-3,2] => [2,1]
=> [1]
=> 1
[1,4,-2,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,4,-2,-3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-4,2,3] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,2,-3] => [1,-4,-3,-2] => [2,1]
=> [1]
=> 1
[1,-4,-2,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,4,2,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,-4,2,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,-2] => [1,-2,4,3] => [2,1]
=> [1]
=> 1
[1,4,-3,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-4,-3,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,4,3,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
Description
The total number of irreducible representations contained in the higher Lie character for an integer partition.
Matching statistic: St001900
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
St001900: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001900: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 1
[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]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 1
[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]
=> 1
[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]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> ? = 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 1
[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]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> ? = 0
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> ? = 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> ? = 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 1
[-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]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[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]
=> 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[1,3,-4,-2] => [1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-3,4,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-3,-4,2] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 1
[1,-3,-4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,3,4,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,3,-4,2] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,2,-3] => [1,4,-3,2] => [2,1]
=> [1]
=> 1
[1,4,-2,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,4,-2,-3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-4,2,3] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,2,-3] => [1,-4,-3,-2] => [2,1]
=> [1]
=> 1
[1,-4,-2,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,4,2,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
[-1,-4,2,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,-2] => [1,-2,4,3] => [2,1]
=> [1]
=> 1
[1,4,-3,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-4,-3,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0
[-1,4,3,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0
Description
The number of distinct irreducible representations contained in the higher Lie character for an integer partition.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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.
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