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Matching statistic: St000481
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000481: 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
St000481: 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,-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
[1,2,-3] => [1,2,-3] => [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
[1,3,-2] => [1,-2,3] => [1,1]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> 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
[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
[-2,1,-3] => [-2,-1,-3] => [2]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> 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
[-3,1,-2] => [-3,-2,-1] => [2]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> 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
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [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
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000183
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
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
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: 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 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: St000480
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ 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
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000480: Integer partitions ⟶ 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]
=> [2]
=> 1
[1,-2] => [1,-2] => [1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> [1,1]
=> 0
[2,-1] => [-1,2] => [1]
=> [1]
=> 0
[-2,1] => [-2,-1] => [2]
=> [1,1]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [3]
=> 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [2]
=> 1
[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]
=> [2,1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [2]
=> 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [2,1]
=> 1
[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]
=> [2,1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> [1,1]
=> 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]
=> [2,1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> [1,1]
=> 0
[2,3,1] => [3,2,1] => [2,1]
=> [2,1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [2]
=> 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [2,1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> [1]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> [2,1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> [1,1]
=> 0
[3,1,2] => [3,2,1] => [2,1]
=> [2,1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> [1,1]
=> 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]
=> [2,1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> [1,1]
=> 0
[3,2,1] => [3,2,1] => [2,1]
=> [2,1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> [1,1]
=> 0
[3,-2,1] => [-2,-1,3] => [2,1]
=> [2,1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> [1]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> [2,1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> [1,1]
=> 0
[-3,-2,1] => [-2,-1,-3] => [2]
=> [1,1]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [3]
=> 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [2]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [2]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [2]
=> 1
[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]
=> [2]
=> 1
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: St000897
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
St000897: 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
St000897: 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 different multiplicities of parts of an integer partition.
Matching statistic: St000659
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●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
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
=> [1,0]
=> ? = 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1]
=> [1,0]
=> ? = 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-3,-4,-2] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[1,-4,-2,3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[1,-4,-3,-2] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[2,-1,3,-4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,-3,4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,-3,-4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,4,-3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,-4,3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,-4,-3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-3,-1,4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-3,-1,-4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-3,4,-1] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-3,-4,-1] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-2,3,4,-1] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[2,-4,-1,3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-4,-1,-3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-4,-3,-1] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-2,4,3,-1] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[3,-1,2,-4] => [-1,-2,3,-4] => [1]
=> [1,0]
=> ? = 0
[3,-1,-2,4] => [-1,-2,3,-4] => [1]
=> [1,0]
=> ? = 0
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St001031
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●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
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
=> [1,0]
=> ? = 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> [1,0,1,0]
=> 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> [1,0,1,0]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1]
=> [1,0]
=> ? = 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-3,-4,-2] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[1,-4,-2,3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[1,-4,-3,-2] => [1,-2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[2,-1,3,-4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,-3,4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,-3,-4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,4,-3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,-4,3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-1,-4,-3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-3,-1,4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-3,-1,-4] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-3,4,-1] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-3,-4,-1] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-2,3,4,-1] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[2,-4,-1,3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-4,-1,-3] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[2,-4,-3,-1] => [-1,2,-3,-4] => [1]
=> [1,0]
=> ? = 0
[-2,4,3,-1] => [-1,-2,-3,4] => [1]
=> [1,0]
=> ? = 0
[3,-1,2,-4] => [-1,-2,3,-4] => [1]
=> [1,0]
=> ? = 0
[3,-1,-2,4] => [-1,-2,3,-4] => [1]
=> [1,0]
=> ? = 0
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St000781
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
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
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 proper colouring schemes 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 is the number of distinct such integer partitions that occur.
Matching statistic: St001901
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
St001901: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
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 largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St001934
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
St001934: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
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 monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St000205
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
St000205: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1]
=> []
=> ? = 0 - 1
[1,2] => [1,2] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-2] => [1,-2] => [1]
=> []
=> ? = 0 - 1
[2,1] => [2,1] => [2]
=> []
=> ? = 0 - 1
[2,-1] => [-1,2] => [1]
=> []
=> ? = 0 - 1
[-2,1] => [-2,-1] => [2]
=> []
=> ? = 0 - 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-2,3] => [1,-2,-3] => [1]
=> []
=> ? = 0 - 1
[1,-2,-3] => [1,-2,-3] => [1]
=> []
=> ? = 0 - 1
[-1,2,3] => [-1,-2,3] => [1]
=> []
=> ? = 0 - 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-3,2] => [1,-3,-2] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,-3,-2] => [1,-2,-3] => [1]
=> []
=> ? = 0 - 1
[-1,3,2] => [-1,-2,3] => [1]
=> []
=> ? = 0 - 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [2]
=> []
=> ? = 0 - 1
[2,-1,3] => [-1,2,-3] => [1]
=> []
=> ? = 0 - 1
[2,-1,-3] => [-1,2,-3] => [1]
=> []
=> ? = 0 - 1
[-2,1,3] => [-2,-1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> []
=> ? = 0 - 1
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1,1]
=> [1]
=> 0 = 1 - 1
[2,-3,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0 = 1 - 1
[2,-3,-1] => [-1,2,-3] => [1]
=> []
=> ? = 0 - 1
[-2,3,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0 - 1
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [2]
=> []
=> ? = 0 - 1
[3,-1,2] => [-1,-2,3] => [1]
=> []
=> ? = 0 - 1
[3,-1,-2] => [-1,-2,3] => [1]
=> []
=> ? = 0 - 1
[-3,1,2] => [-3,2,-1] => [2,1]
=> [1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> []
=> ? = 0 - 1
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [2]
=> []
=> ? = 0 - 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[3,-2,-1] => [-1,-2,3] => [1]
=> []
=> ? = 0 - 1
[-3,2,1] => [-3,2,-1] => [2,1]
=> [1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> []
=> ? = 0 - 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> []
=> ? = 0 - 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0 - 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[1,-2,-4,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[1,-2,-4,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[-1,2,4,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0 - 1
[-1,2,4,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0 - 1
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0 - 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[1,-3,-2,-4] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
=> [1]
=> 0 = 1 - 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1]
=> []
=> ? = 0 - 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,-4,-2] => [1,-2,3,-4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-3,4,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[1,-3,-4,2] => [1,-3,-2,-4] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,-3,-4,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[-1,3,4,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0 - 1
[-1,3,4,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0 - 1
[-1,3,-4,2] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0 - 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,-3] => [1,4,-3,2] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,4,-2,3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,4,-2,-3] => [1,-2,-3,4] => [1,1]
=> [1]
=> 0 = 1 - 1
[1,-4,2,3] => [1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,-4,2,-3] => [1,-4,-3,-2] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,-4,-2,3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[1,-4,-2,-3] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[-1,4,2,3] => [-1,-2,4,3] => [2]
=> []
=> ? = 0 - 1
[-1,4,2,-3] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0 - 1
[-1,-4,2,3] => [-1,-2,-4,-3] => [2]
=> []
=> ? = 0 - 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,-2] => [1,-2,4,3] => [2,1]
=> [1]
=> 0 = 1 - 1
[1,4,-3,2] => [1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,-4,-3,-2] => [1,-2,-3,-4] => [1]
=> []
=> ? = 0 - 1
[-1,4,3,2] => [-1,-2,4,3] => [2]
=> []
=> ? = 0 - 1
[-1,4,3,-2] => [-1,-2,-3,4] => [1]
=> []
=> ? = 0 - 1
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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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