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Matching statistic: St000644
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000644: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000644: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> 1 = 2 - 1
[1,-2] => [1,-2] => [1]
=> 1 = 2 - 1
[-1,2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[-1,-2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[2,-1] => [-1,2] => [1]
=> 1 = 2 - 1
[-2,-1] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1]
=> 1 = 2 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
=> 1 = 2 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
=> 1 = 2 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
=> 1 = 2 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
=> 1 = 2 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> 1 = 2 - 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
=> 1 = 2 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> 1 = 2 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 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 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [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] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
Description
The number of graphs with given frequency partition.
The frequency partition of a graph on $n$ vertices is the partition obtained from its degree sequence by recording and sorting the frequencies of the numbers that occur.
For example, the complete graph on $n$ vertices has frequency partition $(n)$. The path on $n$ vertices has frequency partition $(n-2,2)$, because its degree sequence is $(2,\dots,2,1,1)$. The star graph on $n$ vertices has frequency partition is $(n-1, 1)$, because its degree sequence is $(n-1,1,\dots,1)$.
There are two graphs having frequency partition $(2,1)$: the path and an edge together with an isolated vertex.
Matching statistic: St001121
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001121: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001121: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> 1 = 2 - 1
[1,-2] => [1,-2] => [1]
=> 1 = 2 - 1
[-1,2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[-1,-2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[2,-1] => [-1,2] => [1]
=> 1 = 2 - 1
[-2,-1] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1]
=> 1 = 2 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
=> 1 = 2 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
=> 1 = 2 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
=> 1 = 2 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
=> 1 = 2 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> 1 = 2 - 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
=> 1 = 2 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> 1 = 2 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 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 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [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] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
Description
The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^\lambda$.
Matching statistic: St001122
(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
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> 1 = 2 - 1
[1,-2] => [1,-2] => [1]
=> 1 = 2 - 1
[-1,2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[-1,-2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[2,-1] => [-1,2] => [1]
=> 1 = 2 - 1
[-2,-1] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1]
=> 1 = 2 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
=> 1 = 2 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
=> 1 = 2 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
=> 1 = 2 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
=> 1 = 2 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> 1 = 2 - 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
=> 1 = 2 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> 1 = 2 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 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 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [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] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St001484
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> 1 = 2 - 1
[1,-2] => [1,-2] => [1]
=> 1 = 2 - 1
[-1,2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[-1,-2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[2,-1] => [-1,2] => [1]
=> 1 = 2 - 1
[-2,-1] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1]
=> 1 = 2 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
=> 1 = 2 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
=> 1 = 2 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
=> 1 = 2 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
=> 1 = 2 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> 1 = 2 - 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
=> 1 = 2 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> 1 = 2 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 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 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [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] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St001525
(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
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> 1 = 2 - 1
[1,-2] => [1,-2] => [1]
=> 1 = 2 - 1
[-1,2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[-1,-2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[2,-1] => [-1,2] => [1]
=> 1 = 2 - 1
[-2,-1] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1]
=> 1 = 2 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
=> 1 = 2 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
=> 1 = 2 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
=> 1 = 2 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
=> 1 = 2 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> 1 = 2 - 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
=> 1 = 2 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> 1 = 2 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 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 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [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] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
Description
The number of symmetric hooks on the diagonal of a partition.
Matching statistic: St001593
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> 1 = 2 - 1
[1,-2] => [1,-2] => [1]
=> 1 = 2 - 1
[-1,2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[-1,-2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[2,-1] => [-1,2] => [1]
=> 1 = 2 - 1
[-2,-1] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1]
=> 1 = 2 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
=> 1 = 2 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
=> 1 = 2 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
=> 1 = 2 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
=> 1 = 2 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> 1 = 2 - 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
=> 1 = 2 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> 1 = 2 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 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 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [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] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
Description
This is the number of standard Young tableaux of the given shifted shape.
For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e.,
$$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$
In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
Matching statistic: St001606
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> 1 = 2 - 1
[1,-2] => [1,-2] => [1]
=> 1 = 2 - 1
[-1,2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[-1,-2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[2,-1] => [-1,2] => [1]
=> 1 = 2 - 1
[-2,-1] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1]
=> 1 = 2 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
=> 1 = 2 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
=> 1 = 2 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
=> 1 = 2 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
=> 1 = 2 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> 1 = 2 - 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
=> 1 = 2 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> 1 = 2 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 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 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [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] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions.
Matching statistic: St001939
(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
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> 1 = 2 - 1
[1,-2] => [1,-2] => [1]
=> 1 = 2 - 1
[-1,2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[-1,-2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[2,-1] => [-1,2] => [1]
=> 1 = 2 - 1
[-2,-1] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1]
=> 1 = 2 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
=> 1 = 2 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
=> 1 = 2 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
=> 1 = 2 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
=> 1 = 2 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> 1 = 2 - 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
=> 1 = 2 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> 1 = 2 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 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 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [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] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
Description
The number of parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001940
(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
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> 1 = 2 - 1
[1,-2] => [1,-2] => [1]
=> 1 = 2 - 1
[-1,2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[-1,-2] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[2,-1] => [-1,2] => [1]
=> 1 = 2 - 1
[-2,-1] => [-1,-2] => [1,1]
=> 0 = 1 - 1
[1,2,-3] => [1,2,-3] => [1]
=> 1 = 2 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
=> 1 = 2 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
=> 0 = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
=> 1 = 2 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
=> 1 = 2 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
=> 0 = 1 - 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
=> 1 = 2 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> 1 = 2 - 1
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
=> 1 = 2 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
=> 0 = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> 1 = 2 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
=> 1 = 2 - 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 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 - 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [1,-2,-3,4] => [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] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St000445
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [1]
=> [1,0,1,0]
=> 2
[1,-2] => [1,-2] => [1]
=> [1,0,1,0]
=> 2
[-1,2] => [-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-1,-2] => [-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,-1] => [-1,2] => [1]
=> [1,0,1,0]
=> 2
[-2,-1] => [-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,-3] => [1,2,-3] => [1]
=> [1,0,1,0]
=> 2
[1,-2,3] => [1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,-2,-3] => [1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-1,2,3] => [-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,-2] => [1,-2,3] => [1]
=> [1,0,1,0]
=> 2
[1,-3,-2] => [1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-1,3,2] => [-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,-3] => [2,1,-3] => [1]
=> [1,0,1,0]
=> 2
[2,-1,3] => [-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,-1,-3] => [-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-2,1,-3] => [-2,-1,-3] => [1]
=> [1,0,1,0]
=> 2
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,3,-1] => [-1,2,3] => [1]
=> [1,0,1,0]
=> 2
[2,-3,-1] => [-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[-2,-3,1] => [-2,-1,-3] => [1]
=> [1,0,1,0]
=> 2
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,-2] => [3,-2,1] => [1]
=> [1,0,1,0]
=> 2
[3,-1,2] => [-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,-1,-2] => [-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-3,1,-2] => [-3,-2,-1] => [1]
=> [1,0,1,0]
=> 2
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,-1] => [-1,3,2] => [1]
=> [1,0,1,0]
=> 2
[3,-2,-1] => [-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-3,2,-1] => [-1,-3,-2] => [1]
=> [1,0,1,0]
=> 2
[-3,-2,1] => [-2,-1,-3] => [1]
=> [1,0,1,0]
=> 2
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,3,-4] => [1,2,3,-4] => [1]
=> [1,0,1,0]
=> 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
Description
The number of rises of length 1 of a Dyck path.
The following 115 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000617The number of global maxima of a Dyck path. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000759The smallest missing part in an integer partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001523The degree of symmetry of a Dyck path. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000475The number of parts equal to 1 in a partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001561The value of the elementary symmetric function evaluated at 1. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000981The length of the longest zigzag subpath. 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. 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. St001568The smallest positive integer that does not appear twice in the partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000369The dinv deficit of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000567The sum of the products of all pairs of parts. St000661The number of rises of length 3 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. St000931The number of occurrences of the pattern UUU in 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. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. 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. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. 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. 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. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.
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