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Matching statistic: St001099
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001099: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001099: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
=> 1
[2,-1] => [2]
=> 0
[-2,1] => [2]
=> 0
[1,-2,-3] => [1,1]
=> 1
[-1,2,-3] => [1,1]
=> 1
[-1,-2,3] => [1,1]
=> 1
[-1,-2,-3] => [1,1,1]
=> 3
[1,3,-2] => [2]
=> 0
[1,-3,2] => [2]
=> 0
[-1,3,-2] => [2,1]
=> 2
[-1,-3,2] => [2,1]
=> 2
[2,-1,3] => [2]
=> 0
[2,-1,-3] => [2,1]
=> 2
[-2,1,3] => [2]
=> 0
[-2,1,-3] => [2,1]
=> 2
[2,3,-1] => [3]
=> 0
[2,-3,1] => [3]
=> 0
[-2,3,1] => [3]
=> 0
[-2,-3,-1] => [3]
=> 0
[3,1,-2] => [3]
=> 0
[3,-1,2] => [3]
=> 0
[-3,1,2] => [3]
=> 0
[-3,-1,-2] => [3]
=> 0
[3,2,-1] => [2]
=> 0
[3,-2,-1] => [2,1]
=> 2
[-3,2,1] => [2]
=> 0
[-3,-2,1] => [2,1]
=> 2
[1,2,-3,-4] => [1,1]
=> 1
[1,-2,3,-4] => [1,1]
=> 1
[1,-2,-3,4] => [1,1]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> 3
[-1,2,3,-4] => [1,1]
=> 1
[-1,2,-3,4] => [1,1]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> 3
[-1,-2,3,4] => [1,1]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> 3
[-1,-2,-3,4] => [1,1,1]
=> 3
[-1,-2,-3,-4] => [1,1,1,1]
=> 15
[1,2,4,-3] => [2]
=> 0
[1,2,-4,3] => [2]
=> 0
[1,-2,4,-3] => [2,1]
=> 2
[1,-2,-4,3] => [2,1]
=> 2
[-1,2,4,-3] => [2,1]
=> 2
[-1,2,-4,3] => [2,1]
=> 2
[-1,-2,4,3] => [1,1]
=> 1
[-1,-2,4,-3] => [2,1,1]
=> 12
[-1,-2,-4,3] => [2,1,1]
=> 12
[-1,-2,-4,-3] => [1,1]
=> 1
[1,3,-2,4] => [2]
=> 0
[1,3,-2,-4] => [2,1]
=> 2
Description
The 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.
For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1].
This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for leaf labelled binary trees, with generating function $f(x) = 1-\sqrt{1-2x}$, see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.
Matching statistic: St001491
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 19%●distinct values known / distinct values provided: 5%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 19%●distinct values known / distinct values provided: 5%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[2,-1] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-2,1] => [2]
=> [1,0,1,0]
=> 1010 => 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 3
[1,3,-2] => [2]
=> [1,0,1,0]
=> 1010 => 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[2,-1,3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> 1010 => 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 3
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 3
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 3
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 15
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 12
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 12
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> 1010 => 0
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[1,-3,2,4] => [2]
=> [1,0,1,0]
=> 1010 => 0
[1,-3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-1,3,2,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 12
[-1,-3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 12
[-1,-3,-2,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[-1,3,4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 6
[-1,3,-4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 6
[-1,-3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 6
[-1,-3,-4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 6
[1,4,2,-3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[1,4,-2,3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[1,-4,2,3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[1,-4,-2,-3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 0
[-1,4,2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 6
[-1,4,-2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 6
[-1,-4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 6
[-1,-4,-2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 6
[1,4,3,-2] => [2]
=> [1,0,1,0]
=> 1010 => 0
[1,4,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[1,-4,3,2] => [2]
=> [1,0,1,0]
=> 1010 => 0
[1,-4,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2
[-1,4,-3,2] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-1,-4,-3,-2] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[2,1,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[2,-1,3,4] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-2,1,3,4] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-2,-1,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[2,1,4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[2,1,-4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[2,-1,4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[2,-1,-4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-2,1,4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-2,1,-4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-2,-1,4,-3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-2,-1,-4,3] => [2]
=> [1,0,1,0]
=> 1010 => 0
[3,2,-1,4] => [2]
=> [1,0,1,0]
=> 1010 => 0
[3,-2,1,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[-3,2,1,4] => [2]
=> [1,0,1,0]
=> 1010 => 0
[-3,-2,-1,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1
[3,4,1,-2] => [2]
=> [1,0,1,0]
=> 1010 => 0
[3,4,-1,2] => [2]
=> [1,0,1,0]
=> 1010 => 0
[3,-4,1,2] => [2]
=> [1,0,1,0]
=> 1010 => 0
[3,-4,-1,-2] => [2]
=> [1,0,1,0]
=> 1010 => 0
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000512
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 1
[2,-1] => []
=> ?
=> ?
=> ? = 0
[-2,1] => []
=> ?
=> ?
=> ? = 0
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 3
[1,3,-2] => [1]
=> []
=> ?
=> ? = 0
[1,-3,2] => [1]
=> []
=> ?
=> ? = 0
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2
[2,-1,3] => [1]
=> []
=> ?
=> ? = 0
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2
[-2,1,3] => [1]
=> []
=> ?
=> ? = 0
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2
[2,3,-1] => []
=> ?
=> ?
=> ? = 0
[2,-3,1] => []
=> ?
=> ?
=> ? = 0
[-2,3,1] => []
=> ?
=> ?
=> ? = 0
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 0
[3,1,-2] => []
=> ?
=> ?
=> ? = 0
[3,-1,2] => []
=> ?
=> ?
=> ? = 0
[-3,1,2] => []
=> ?
=> ?
=> ? = 0
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 0
[3,2,-1] => [1]
=> []
=> ?
=> ? = 0
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2
[-3,2,1] => [1]
=> []
=> ?
=> ? = 0
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 15
[1,2,4,-3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,2,-4,3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 1
[1,3,-2,4] => [1,1]
=> [1]
=> []
=> ? = 0
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2
[1,2,3,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,5,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,5,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,5,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,5,-1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[-2,1,3,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,7,1,4,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,7,3,1,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,7,3,4,1,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,7,3,4,5,1,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,7,2,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,7,4,2,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,7,4,5,2,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,7,1,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,7,4,1,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,7,4,5,1,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,7,3,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,7,5,3,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,7,2,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,7,5,2,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,3,7,1,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,7,5,1,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,7,4,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,7,3,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,3,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,7,2,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,2,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,7,1,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,1,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,7,5,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,5,4,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,5,3,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,5,2,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,5,1,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,6,5,8,7,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,2,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,-4,3,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,-5,4,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,-6,5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,6,5,8,7,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,5,4,1,8,7,-6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,8,7,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1,8,7,-6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000936
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 1
[2,-1] => []
=> ?
=> ?
=> ? = 0
[-2,1] => []
=> ?
=> ?
=> ? = 0
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 3
[1,3,-2] => [1]
=> []
=> ?
=> ? = 0
[1,-3,2] => [1]
=> []
=> ?
=> ? = 0
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2
[2,-1,3] => [1]
=> []
=> ?
=> ? = 0
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2
[-2,1,3] => [1]
=> []
=> ?
=> ? = 0
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2
[2,3,-1] => []
=> ?
=> ?
=> ? = 0
[2,-3,1] => []
=> ?
=> ?
=> ? = 0
[-2,3,1] => []
=> ?
=> ?
=> ? = 0
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 0
[3,1,-2] => []
=> ?
=> ?
=> ? = 0
[3,-1,2] => []
=> ?
=> ?
=> ? = 0
[-3,1,2] => []
=> ?
=> ?
=> ? = 0
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 0
[3,2,-1] => [1]
=> []
=> ?
=> ? = 0
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2
[-3,2,1] => [1]
=> []
=> ?
=> ? = 0
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 15
[1,2,4,-3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,2,-4,3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 1
[1,3,-2,4] => [1,1]
=> [1]
=> []
=> ? = 0
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2
[1,2,3,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,5,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,5,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,5,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,5,-1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[-2,1,3,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,7,1,4,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,7,3,1,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,7,3,4,1,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,7,3,4,5,1,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,7,2,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,7,4,2,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,7,4,5,2,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,7,1,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,7,4,1,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,7,4,5,1,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,7,3,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,7,5,3,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,7,2,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,7,5,2,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,3,7,1,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,7,5,1,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,7,4,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,7,3,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,3,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,7,2,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,2,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,7,1,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,1,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,7,5,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,5,4,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,5,3,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,5,2,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,5,1,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,6,5,8,7,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,2,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,-4,3,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,-5,4,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,-6,5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,6,5,8,7,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,5,4,1,8,7,-6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,8,7,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1,8,7,-6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of even values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$.
It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St000938
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 1
[2,-1] => []
=> ?
=> ?
=> ? = 0
[-2,1] => []
=> ?
=> ?
=> ? = 0
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 3
[1,3,-2] => [1]
=> []
=> ?
=> ? = 0
[1,-3,2] => [1]
=> []
=> ?
=> ? = 0
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2
[2,-1,3] => [1]
=> []
=> ?
=> ? = 0
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2
[-2,1,3] => [1]
=> []
=> ?
=> ? = 0
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2
[2,3,-1] => []
=> ?
=> ?
=> ? = 0
[2,-3,1] => []
=> ?
=> ?
=> ? = 0
[-2,3,1] => []
=> ?
=> ?
=> ? = 0
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 0
[3,1,-2] => []
=> ?
=> ?
=> ? = 0
[3,-1,2] => []
=> ?
=> ?
=> ? = 0
[-3,1,2] => []
=> ?
=> ?
=> ? = 0
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 0
[3,2,-1] => [1]
=> []
=> ?
=> ? = 0
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2
[-3,2,1] => [1]
=> []
=> ?
=> ? = 0
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 15
[1,2,4,-3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,2,-4,3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 1
[1,3,-2,4] => [1,1]
=> [1]
=> []
=> ? = 0
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2
[1,2,3,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,5,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,5,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,5,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,5,-1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[-2,1,3,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,7,1,4,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,7,3,1,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,7,3,4,1,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,7,3,4,5,1,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,7,2,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,7,4,2,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,7,4,5,2,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,7,1,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,7,4,1,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,7,4,5,1,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,7,3,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,7,5,3,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,7,2,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,7,5,2,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,3,7,1,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,7,5,1,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,7,4,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,7,3,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,3,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,7,2,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,2,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,7,1,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,1,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,7,5,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,5,4,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,5,3,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,5,2,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,5,1,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,6,5,8,7,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,2,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,-4,3,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,-5,4,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,-6,5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,6,5,8,7,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,5,4,1,8,7,-6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,8,7,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1,8,7,-6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000940
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 1
[2,-1] => []
=> ?
=> ?
=> ? = 0
[-2,1] => []
=> ?
=> ?
=> ? = 0
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 3
[1,3,-2] => [1]
=> []
=> ?
=> ? = 0
[1,-3,2] => [1]
=> []
=> ?
=> ? = 0
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2
[2,-1,3] => [1]
=> []
=> ?
=> ? = 0
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2
[-2,1,3] => [1]
=> []
=> ?
=> ? = 0
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2
[2,3,-1] => []
=> ?
=> ?
=> ? = 0
[2,-3,1] => []
=> ?
=> ?
=> ? = 0
[-2,3,1] => []
=> ?
=> ?
=> ? = 0
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 0
[3,1,-2] => []
=> ?
=> ?
=> ? = 0
[3,-1,2] => []
=> ?
=> ?
=> ? = 0
[-3,1,2] => []
=> ?
=> ?
=> ? = 0
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 0
[3,2,-1] => [1]
=> []
=> ?
=> ? = 0
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2
[-3,2,1] => [1]
=> []
=> ?
=> ? = 0
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 15
[1,2,4,-3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,2,-4,3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 1
[1,3,-2,4] => [1,1]
=> [1]
=> []
=> ? = 0
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2
[1,2,3,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,5,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,5,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,5,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,5,-1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[-2,1,3,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,7,1,4,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,7,3,1,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,7,3,4,1,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,7,3,4,5,1,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,7,2,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,7,4,2,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,7,4,5,2,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,7,1,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,7,4,1,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,7,4,5,1,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,7,3,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,7,5,3,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,7,2,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,7,5,2,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,3,7,1,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,7,5,1,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,7,4,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,7,3,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,3,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,7,2,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,2,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,7,1,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,1,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,7,5,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,5,4,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,5,3,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,5,2,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,5,1,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,6,5,8,7,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,2,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,-4,3,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,-5,4,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,-6,5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,6,5,8,7,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,5,4,1,8,7,-6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,8,7,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1,8,7,-6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
Matching statistic: St000941
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 1
[2,-1] => []
=> ?
=> ?
=> ? = 0
[-2,1] => []
=> ?
=> ?
=> ? = 0
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 3
[1,3,-2] => [1]
=> []
=> ?
=> ? = 0
[1,-3,2] => [1]
=> []
=> ?
=> ? = 0
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2
[2,-1,3] => [1]
=> []
=> ?
=> ? = 0
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2
[-2,1,3] => [1]
=> []
=> ?
=> ? = 0
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2
[2,3,-1] => []
=> ?
=> ?
=> ? = 0
[2,-3,1] => []
=> ?
=> ?
=> ? = 0
[-2,3,1] => []
=> ?
=> ?
=> ? = 0
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 0
[3,1,-2] => []
=> ?
=> ?
=> ? = 0
[3,-1,2] => []
=> ?
=> ?
=> ? = 0
[-3,1,2] => []
=> ?
=> ?
=> ? = 0
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 0
[3,2,-1] => [1]
=> []
=> ?
=> ? = 0
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2
[-3,2,1] => [1]
=> []
=> ?
=> ? = 0
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 15
[1,2,4,-3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,2,-4,3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 1
[1,3,-2,4] => [1,1]
=> [1]
=> []
=> ? = 0
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2
[1,2,3,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,5,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,5,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,5,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,5,-1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[-2,1,3,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,7,1,4,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,7,3,1,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,7,3,4,1,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,7,3,4,5,1,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,7,2,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,7,4,2,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,7,4,5,2,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,7,1,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,7,4,1,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,7,4,5,1,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,7,3,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,7,5,3,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,7,2,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,7,5,2,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,3,7,1,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,7,5,1,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,7,4,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,7,3,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,3,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,7,2,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,2,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,7,1,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,1,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,7,5,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,5,4,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,5,3,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,5,2,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,5,1,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,6,5,8,7,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,2,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,-4,3,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,-5,4,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,-6,5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,6,5,8,7,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,5,4,1,8,7,-6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,8,7,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1,8,7,-6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of characters of the symmetric group whose value on the partition is even.
Matching statistic: St001124
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 1
[2,-1] => []
=> ?
=> ?
=> ? = 0
[-2,1] => []
=> ?
=> ?
=> ? = 0
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 3
[1,3,-2] => [1]
=> []
=> ?
=> ? = 0
[1,-3,2] => [1]
=> []
=> ?
=> ? = 0
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2
[2,-1,3] => [1]
=> []
=> ?
=> ? = 0
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2
[-2,1,3] => [1]
=> []
=> ?
=> ? = 0
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2
[2,3,-1] => []
=> ?
=> ?
=> ? = 0
[2,-3,1] => []
=> ?
=> ?
=> ? = 0
[-2,3,1] => []
=> ?
=> ?
=> ? = 0
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 0
[3,1,-2] => []
=> ?
=> ?
=> ? = 0
[3,-1,2] => []
=> ?
=> ?
=> ? = 0
[-3,1,2] => []
=> ?
=> ?
=> ? = 0
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 0
[3,2,-1] => [1]
=> []
=> ?
=> ? = 0
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2
[-3,2,1] => [1]
=> []
=> ?
=> ? = 0
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 3
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 15
[1,2,4,-3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,2,-4,3] => [1,1]
=> [1]
=> []
=> ? = 0
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 12
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 1
[1,3,-2,4] => [1,1]
=> [1]
=> []
=> ? = 0
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2
[1,2,3,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,5,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,5,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,5,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,5,-1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[-2,1,3,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,7,1,4,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,7,3,1,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,7,3,4,1,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,7,3,4,5,1,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,7,2,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,7,4,2,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,7,4,5,2,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,7,1,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,7,4,1,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,7,4,5,1,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,7,3,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,7,5,3,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,7,2,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,7,5,2,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,3,7,1,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,7,5,1,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,7,4,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,7,3,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,3,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,3,4,7,2,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,2,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,2,3,4,7,1,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,1,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,7,5,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,7,5,4,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,7,4,5,3,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,7,3,4,5,2,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[7,2,3,4,5,1,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,6,5,8,7,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,-3,2,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,-4,3,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,-5,4,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,-6,5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,6,5,8,7,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,5,4,1,8,7,-6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,6,5,4,3,8,7,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4,3,2,1,8,7,-6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The multiplicity of the standard 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 $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000284
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 1 + 1
[2,-1] => []
=> ?
=> ?
=> ? = 0 + 1
[-2,1] => []
=> ?
=> ?
=> ? = 0 + 1
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 1 + 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 1 + 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 1 + 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 3 + 1
[1,3,-2] => [1]
=> []
=> ?
=> ? = 0 + 1
[1,-3,2] => [1]
=> []
=> ?
=> ? = 0 + 1
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2 + 1
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2 + 1
[2,-1,3] => [1]
=> []
=> ?
=> ? = 0 + 1
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2 + 1
[-2,1,3] => [1]
=> []
=> ?
=> ? = 0 + 1
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2 + 1
[2,3,-1] => []
=> ?
=> ?
=> ? = 0 + 1
[2,-3,1] => []
=> ?
=> ?
=> ? = 0 + 1
[-2,3,1] => []
=> ?
=> ?
=> ? = 0 + 1
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 0 + 1
[3,1,-2] => []
=> ?
=> ?
=> ? = 0 + 1
[3,-1,2] => []
=> ?
=> ?
=> ? = 0 + 1
[-3,1,2] => []
=> ?
=> ?
=> ? = 0 + 1
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 0 + 1
[3,2,-1] => [1]
=> []
=> ?
=> ? = 0 + 1
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2 + 1
[-3,2,1] => [1]
=> []
=> ?
=> ? = 0 + 1
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2 + 1
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 3 + 1
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 3 + 1
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 3 + 1
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 3 + 1
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 15 + 1
[1,2,4,-3] => [1,1]
=> [1]
=> []
=> ? = 0 + 1
[1,2,-4,3] => [1,1]
=> [1]
=> []
=> ? = 0 + 1
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2 + 1
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2 + 1
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2 + 1
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2 + 1
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 1 + 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 12 + 1
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 12 + 1
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 1 + 1
[1,3,-2,4] => [1,1]
=> [1]
=> []
=> ? = 0 + 1
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2 + 1
[1,2,3,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,6,5,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,6,4,5,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,3,4,5,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,2,3,4,5,-1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-2,1,3,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,7,1,4,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,7,3,1,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[5,7,3,4,1,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,7,3,4,5,1,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,7,2,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5,7,4,2,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,7,4,5,2,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,2,7,1,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[5,2,7,4,1,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,2,7,4,5,1,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,5,7,3,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,6,7,5,3,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5,3,7,2,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,3,7,5,2,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[5,2,3,7,1,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,2,3,7,5,1,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,6,7,4,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,6,4,7,3,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,7,4,3,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,3,4,7,2,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,7,3,4,2,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,2,3,4,7,1,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,2,3,4,1,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4,7,5,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,7,5,4,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,7,4,5,3,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,7,3,4,5,2,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,2,3,4,5,1,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,3,6,5,8,7,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,-3,2,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,-4,3,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,-5,4,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4,-6,5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,2,1,6,5,8,7,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,2,5,4,1,8,7,-6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,5,4,3,8,7,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[5,4,3,2,1,8,7,-6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
Description
The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000510
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 3%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 1 + 1
[2,-1] => []
=> ?
=> ?
=> ? = 0 + 1
[-2,1] => []
=> ?
=> ?
=> ? = 0 + 1
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 1 + 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 1 + 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 1 + 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 3 + 1
[1,3,-2] => [1]
=> []
=> ?
=> ? = 0 + 1
[1,-3,2] => [1]
=> []
=> ?
=> ? = 0 + 1
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2 + 1
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2 + 1
[2,-1,3] => [1]
=> []
=> ?
=> ? = 0 + 1
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2 + 1
[-2,1,3] => [1]
=> []
=> ?
=> ? = 0 + 1
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2 + 1
[2,3,-1] => []
=> ?
=> ?
=> ? = 0 + 1
[2,-3,1] => []
=> ?
=> ?
=> ? = 0 + 1
[-2,3,1] => []
=> ?
=> ?
=> ? = 0 + 1
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 0 + 1
[3,1,-2] => []
=> ?
=> ?
=> ? = 0 + 1
[3,-1,2] => []
=> ?
=> ?
=> ? = 0 + 1
[-3,1,2] => []
=> ?
=> ?
=> ? = 0 + 1
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 0 + 1
[3,2,-1] => [1]
=> []
=> ?
=> ? = 0 + 1
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2 + 1
[-3,2,1] => [1]
=> []
=> ?
=> ? = 0 + 1
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2 + 1
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 3 + 1
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 3 + 1
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 1 + 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 3 + 1
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 3 + 1
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 15 + 1
[1,2,4,-3] => [1,1]
=> [1]
=> []
=> ? = 0 + 1
[1,2,-4,3] => [1,1]
=> [1]
=> []
=> ? = 0 + 1
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2 + 1
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2 + 1
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2 + 1
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2 + 1
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 1 + 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 12 + 1
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 12 + 1
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 1 + 1
[1,3,-2,4] => [1,1]
=> [1]
=> []
=> ? = 0 + 1
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2 + 1
[1,2,3,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,6,5,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,6,4,5,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,3,4,5,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,2,3,4,5,-1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-2,1,3,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,7,1,4,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,7,3,1,5,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[5,7,3,4,1,6,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,7,3,4,5,1,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,7,2,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5,7,4,2,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,7,4,5,2,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,2,7,1,5,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[5,2,7,4,1,6,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,2,7,4,5,1,-3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,5,7,3,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,6,7,5,3,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5,3,7,2,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,3,7,5,2,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[5,2,3,7,1,6,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,2,3,7,5,1,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,6,7,4,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,6,4,7,3,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,7,4,3,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,3,4,7,2,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,7,3,4,2,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6,2,3,4,7,1,-5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,2,3,4,1,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4,7,5,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,7,5,4,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,7,4,5,3,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,7,3,4,5,2,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[7,2,3,4,5,1,-6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,3,6,5,8,7,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,-3,2,4,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,-4,3,5,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,-5,4,6] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4,-6,5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,2,1,6,5,8,7,-4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,2,5,4,1,8,7,-6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,6,5,4,3,8,7,-2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[5,4,3,2,1,8,7,-6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type.
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