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Your data matches 24 different statistics following compositions of up to 3 maps.
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Matching statistic: St000567

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000567: 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]
 => 6
[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]
 => 5
[-1,-2,-4,3] => [2,1,1]
 => 5
[-1,-2,-4,-3] => [1,1]
 => 1
[1,3,-2,4] => [2]
 => 0
[1,3,-2,-4] => [2,1]
 => 2

Description

The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to

$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$ for a partition

$\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$ , see [1]. This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].

Matching statistic: St000059

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000059: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[-1,-2] => [1,1]
 => [[1],[2]]
 => 1
[2,-1] => [2]
 => [[1,2]]
 => 0
[-2,1] => [2]
 => [[1,2]]
 => 0
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => 1
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => 1
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => 1
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => 3
[1,3,-2] => [2]
 => [[1,2]]
 => 0
[1,-3,2] => [2]
 => [[1,2]]
 => 0
[-1,3,-2] => [2,1]
 => [[1,2],[3]]
 => 2
[-1,-3,2] => [2,1]
 => [[1,2],[3]]
 => 2
[2,-1,3] => [2]
 => [[1,2]]
 => 0
[2,-1,-3] => [2,1]
 => [[1,2],[3]]
 => 2
[-2,1,3] => [2]
 => [[1,2]]
 => 0
[-2,1,-3] => [2,1]
 => [[1,2],[3]]
 => 2
[2,3,-1] => [3]
 => [[1,2,3]]
 => 0
[2,-3,1] => [3]
 => [[1,2,3]]
 => 0
[-2,3,1] => [3]
 => [[1,2,3]]
 => 0
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => 0
[3,1,-2] => [3]
 => [[1,2,3]]
 => 0
[3,-1,2] => [3]
 => [[1,2,3]]
 => 0
[-3,1,2] => [3]
 => [[1,2,3]]
 => 0
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => 0
[3,2,-1] => [2]
 => [[1,2]]
 => 0
[3,-2,-1] => [2,1]
 => [[1,2],[3]]
 => 2
[-3,2,1] => [2]
 => [[1,2]]
 => 0
[-3,-2,1] => [2,1]
 => [[1,2],[3]]
 => 2
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => 1
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => 1
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => 1
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => 3
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => 1
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => 1
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => 3
[-1,-2,3,4] => [1,1]
 => [[1],[2]]
 => 1
[-1,-2,3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => 3
[-1,-2,-3,4] => [1,1,1]
 => [[1],[2],[3]]
 => 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 6
[1,2,4,-3] => [2]
 => [[1,2]]
 => 0
[1,2,-4,3] => [2]
 => [[1,2]]
 => 0
[1,-2,4,-3] => [2,1]
 => [[1,2],[3]]
 => 2
[1,-2,-4,3] => [2,1]
 => [[1,2],[3]]
 => 2
[-1,2,4,-3] => [2,1]
 => [[1,2],[3]]
 => 2
[-1,2,-4,3] => [2,1]
 => [[1,2],[3]]
 => 2
[-1,-2,4,3] => [1,1]
 => [[1],[2]]
 => 1
[-1,-2,4,-3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 5
[-1,-2,-4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 5
[-1,-2,-4,-3] => [1,1]
 => [[1],[2]]
 => 1
[1,3,-2,4] => [2]
 => [[1,2]]
 => 0
[1,3,-2,-4] => [2,1]
 => [[1,2],[3]]
 => 2

Description

The inversion number of a standard tableau as defined by Haglund and Stevens. Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].

Matching statistic: St001541

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001541: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[-1,-2] => [1,1]
 => [2]
 => 1
[2,-1] => [2]
 => [1,1]
 => 0
[-2,1] => [2]
 => [1,1]
 => 0
[1,-2,-3] => [1,1]
 => [2]
 => 1
[-1,2,-3] => [1,1]
 => [2]
 => 1
[-1,-2,3] => [1,1]
 => [2]
 => 1
[-1,-2,-3] => [1,1,1]
 => [3]
 => 3
[1,3,-2] => [2]
 => [1,1]
 => 0
[1,-3,2] => [2]
 => [1,1]
 => 0
[-1,3,-2] => [2,1]
 => [2,1]
 => 2
[-1,-3,2] => [2,1]
 => [2,1]
 => 2
[2,-1,3] => [2]
 => [1,1]
 => 0
[2,-1,-3] => [2,1]
 => [2,1]
 => 2
[-2,1,3] => [2]
 => [1,1]
 => 0
[-2,1,-3] => [2,1]
 => [2,1]
 => 2
[2,3,-1] => [3]
 => [1,1,1]
 => 0
[2,-3,1] => [3]
 => [1,1,1]
 => 0
[-2,3,1] => [3]
 => [1,1,1]
 => 0
[-2,-3,-1] => [3]
 => [1,1,1]
 => 0
[3,1,-2] => [3]
 => [1,1,1]
 => 0
[3,-1,2] => [3]
 => [1,1,1]
 => 0
[-3,1,2] => [3]
 => [1,1,1]
 => 0
[-3,-1,-2] => [3]
 => [1,1,1]
 => 0
[3,2,-1] => [2]
 => [1,1]
 => 0
[3,-2,-1] => [2,1]
 => [2,1]
 => 2
[-3,2,1] => [2]
 => [1,1]
 => 0
[-3,-2,1] => [2,1]
 => [2,1]
 => 2
[1,2,-3,-4] => [1,1]
 => [2]
 => 1
[1,-2,3,-4] => [1,1]
 => [2]
 => 1
[1,-2,-3,4] => [1,1]
 => [2]
 => 1
[1,-2,-3,-4] => [1,1,1]
 => [3]
 => 3
[-1,2,3,-4] => [1,1]
 => [2]
 => 1
[-1,2,-3,4] => [1,1]
 => [2]
 => 1
[-1,2,-3,-4] => [1,1,1]
 => [3]
 => 3
[-1,-2,3,4] => [1,1]
 => [2]
 => 1
[-1,-2,3,-4] => [1,1,1]
 => [3]
 => 3
[-1,-2,-3,4] => [1,1,1]
 => [3]
 => 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [4]
 => 6
[1,2,4,-3] => [2]
 => [1,1]
 => 0
[1,2,-4,3] => [2]
 => [1,1]
 => 0
[1,-2,4,-3] => [2,1]
 => [2,1]
 => 2
[1,-2,-4,3] => [2,1]
 => [2,1]
 => 2
[-1,2,4,-3] => [2,1]
 => [2,1]
 => 2
[-1,2,-4,3] => [2,1]
 => [2,1]
 => 2
[-1,-2,4,3] => [1,1]
 => [2]
 => 1
[-1,-2,4,-3] => [2,1,1]
 => [3,1]
 => 5
[-1,-2,-4,3] => [2,1,1]
 => [3,1]
 => 5
[-1,-2,-4,-3] => [1,1]
 => [2]
 => 1
[1,3,-2,4] => [2]
 => [1,1]
 => 0
[1,3,-2,-4] => [2,1]
 => [2,1]
 => 2

Description

The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.

Matching statistic: St000009

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[-1,-2] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[2,-1] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[-2,1] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[1,-2,-3] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[-1,2,-3] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[-1,-2,3] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[-1,-2,-3] => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
[1,3,-2] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[1,-3,2] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[-1,3,-2] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[-1,-3,2] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[2,-1,3] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[2,-1,-3] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[-2,1,3] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[-2,1,-3] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[2,3,-1] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,-3,1] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[-2,3,1] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[-2,-3,-1] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,1,-2] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,-1,2] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[-3,1,2] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[-3,-1,-2] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,2,-1] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[3,-2,-1] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[-3,2,1] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[-3,-2,1] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[1,2,-3,-4] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[1,-2,3,-4] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[1,-2,-3,4] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[1,-2,-3,-4] => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
[-1,2,3,-4] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[-1,2,-3,4] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[-1,2,-3,-4] => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
[-1,-2,3,4] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[-1,-2,3,-4] => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
[-1,-2,-3,4] => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 6
[1,2,4,-3] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[1,2,-4,3] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[1,-2,4,-3] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[1,-2,-4,3] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[-1,2,4,-3] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[-1,2,-4,3] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[-1,-2,4,3] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[-1,-2,4,-3] => [2,1,1]
 => [3,1]
 => [[1,2,3],[4]]
 => 5
[-1,-2,-4,3] => [2,1,1]
 => [3,1]
 => [[1,2,3],[4]]
 => 5
[-1,-2,-4,-3] => [1,1]
 => [2]
 => [[1,2]]
 => 1
[1,3,-2,4] => [2]
 => [1,1]
 => [[1],[2]]
 => 0
[1,3,-2,-4] => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2

Description

The charge of a standard tableau.

Matching statistic: St000018

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[-1,-2] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[2,-1] => [2]
 => [[1,2]]
 => [1,2] => 0
[-2,1] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[1,3,-2] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,-3,2] => [2]
 => [[1,2]]
 => [1,2] => 0
[-1,3,-2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-1,-3,2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[2,-1,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[2,-1,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-2,1,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[-2,1,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[2,3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[2,-3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-2,3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,-1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-3,1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,2,-1] => [2]
 => [[1,2]]
 => [1,2] => 0
[3,-2,-1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-3,2,1] => [2]
 => [[1,2]]
 => [1,2] => 0
[-3,-2,1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[-1,-2,3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[-1,-2,-3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 6
[1,2,4,-3] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,2,-4,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,-2,4,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[1,-2,-4,3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-1,2,4,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-1,2,-4,3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-1,-2,4,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,4,-3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 5
[-1,-2,-4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 5
[-1,-2,-4,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,3,-2,4] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,3,-2,-4] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2

Description

The number of inversions of a permutation. This equals the minimal number of simple transpositions

$(i,i+1)$ needed to write

$\pi$ . Thus, it is also the Coxeter length of

$\pi$ .

Matching statistic: St000564
(load all 2 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000564: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%

Values

[-1,-2] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[2,-1] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[-2,1] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 3
[1,3,-2] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[1,-3,2] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[-1,3,-2] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[-1,-3,2] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[2,-1,3] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[2,-1,-3] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[-2,1,3] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[-2,1,-3] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[2,3,-1] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 0
[2,-3,1] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 0
[-2,3,1] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 0
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 0
[3,1,-2] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 0
[3,-1,2] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 0
[-3,1,2] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 0
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 0
[3,2,-1] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[3,-2,-1] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[-3,2,1] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[-3,-2,1] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 3
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 3
[-1,-2,3,4] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[-1,-2,3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 3
[-1,-2,-3,4] => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 6
[1,2,4,-3] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[1,2,-4,3] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[1,-2,4,-3] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[1,-2,-4,3] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[-1,2,4,-3] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[-1,2,-4,3] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[-1,-2,4,3] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[-1,-2,4,-3] => [2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 5
[-1,-2,-4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 5
[-1,-2,-4,-3] => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 1
[1,3,-2,4] => [2]
 => [[1,2]]
 => {{1,2}}
 => 0
[1,3,-2,-4] => [2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 2
[-8,-6,-4,-2,1,3,5,7] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 16
[-5,4,-8,-6,-2,1,3,7] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[4,-5,-8,-6,-3,1,2,7] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[3,-8,-5,-6,-4,1,2,7] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[-5,4,-7,3,8,1,2,6] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 16
[6,7,4,5,-8,-2,1,3] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 16
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 24
[-5,4,-6,3,-7,2,-8,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 16
[4,8,-7,-6,3,1,2,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[4,-8,6,-7,-5,-3,1,2] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[-7,-3,2,8,-5,1,4,6] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 17
[7,8,-6,2,3,5,-4,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[4,7,8,-6,2,3,-5,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[-2,1,-4,3,-6,5,-8,7] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 24
[2,-4,-3,1,-6,5,-8,7] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[2,-6,4,-3,-5,1,-8,7] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[2,-8,4,-3,6,-5,-7,1] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[2,-5,-4,6,1,3,-8,7] => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => {{1,2,5},{3,4,8},{6,7}}
 => ? = 21
[-2,1,4,-6,-5,3,-8,7] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[4,-6,1,-3,-5,2,-8,7] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[-2,1,4,-8,6,-5,-7,3] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[4,-8,1,-3,6,-5,-7,2] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[3,5,-8,2,-7,-4,-6,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[-2,1,4,-7,-6,8,3,5] => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => {{1,2,5},{3,4,8},{6,7}}
 => ? = 21
[3,8,-7,-6,-5,2,1,4] => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => ? = 19
[7,8,-3,-6,2,4,-5,1] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 17
[7,8,1,-3,2,-5,4,6] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 16
[-2,1,-4,3,6,-8,-7,5] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[2,-4,-3,1,6,-8,-7,5] => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => {{1,4,5},{2,7,8},{3},{6}}
 => ? = 22
[2,-7,-4,8,-6,5,1,3] => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => {{1,2,5},{3,4,8},{6,7}}
 => ? = 21
[-2,1,6,-8,3,-5,-7,4] => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => ? = 23
[7,8,5,1,-6,2,-4,3] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[7,8,4,-6,2,3,-5,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[2,8,-7,-6,-5,4,1,3] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 17
[5,3,-8,1,-7,-4,-6,2] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[-2,-3,-4,1,-6,5,-8,7] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 20
[-2,-5,4,-3,-6,1,-8,7] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 20
[7,5,2,3,6,-4,-8,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[6,-4,-5,2,1,3,-8,7] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 20
[-2,1,-4,-5,-6,3,-8,7] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 20
[4,-6,-5,3,1,2,-8,7] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 20
[8,-3,-4,-6,-7,2,1,5] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 16
[-2,1,-4,-7,6,-5,-8,3] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 20
[6,-3,-7,-4,1,-5,-8,2] => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => ? = 19
[-5,-3,-6,2,-7,-4,-8,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 16
[-7,3,-5,-8,2,4,-6,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[2,-5,-6,7,1,8,3,4] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 15
[8,-7,2,-4,1,-5,3,6] => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => ? = 19
[-2,1,-4,3,-6,-7,-8,5] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 20
[-2,-3,-4,1,-6,-7,-8,5] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 16

Description

The number of occurrences of the pattern {{1},{2}} in a set partition.

Matching statistic: St001759

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 95%●distinct values known / distinct values provided: 58%

Values

[-1,-2] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[2,-1] => [2]
 => [[1,2]]
 => [1,2] => 0
[-2,1] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[1,3,-2] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,-3,2] => [2]
 => [[1,2]]
 => [1,2] => 0
[-1,3,-2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-1,-3,2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[2,-1,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[2,-1,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-2,1,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[-2,1,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[2,3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[2,-3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-2,3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,-1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-3,1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,2,-1] => [2]
 => [[1,2]]
 => [1,2] => 0
[3,-2,-1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-3,2,1] => [2]
 => [[1,2]]
 => [1,2] => 0
[-3,-2,1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[-1,-2,3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[-1,-2,-3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 6
[1,2,4,-3] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,2,-4,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,-2,4,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[1,-2,-4,3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-1,2,4,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-1,2,-4,3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-1,-2,4,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,4,-3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 5
[-1,-2,-4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 5
[-1,-2,-4,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,3,-2,4] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,3,-2,-4] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[-8,-6,-4,-2,1,3,5,7] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 16
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ? = 6
[5,-3,2,4,8,-7,1,6] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ? = 12
[-5,4,-8,-6,-2,1,3,7] => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ? = 15
[4,-5,-8,-6,-3,1,2,7] => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ? = 15
[3,-8,-5,-6,-4,1,2,7] => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ? = 15
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[-5,4,-7,3,8,1,2,6] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 16
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ? = 12
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ? = 12
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[-7,-6,3,-8,4,-5,1,2] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ? = 12
[6,7,4,5,-8,-2,1,3] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 16
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ? = 12
[4,2,-7,8,-6,1,3,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ? = 12
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ? = 24
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ? = 17
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[-7,2,-4,6,-8,-5,1,3] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ? = 12
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ? = 6
[5,7,-6,4,2,3,-8,1] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 7
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ? = 12
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ? = 6
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ? = 6
[-5,4,-6,3,-7,2,-8,1] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 16
[-7,4,-3,1,2,5,6] => [6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ? = 6
[-2,1,-7,3,4,5,6] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[-2,-7,-6,-5,1,3,4] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[6,-5,2,-7,1,3,4] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[3,1,-7,-6,2,4,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[4,1,-7,-6,-5,2,3] => [4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ? = 14
[2,3,-7,-6,1,4,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[-7,1,-6,5,2,3,4] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[3,4,-7,-2,1,5,6] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[-6,4,-7,-3,-2,1,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[-6,-5,-4,2,-7,1,3] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 10
[3,1,5,2,8,4,6,-7] => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ? = 0
[4,5,1,2,8,3,6,-7] => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ? = 0
[7,4,1,8,2,3,5,-6] => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ? = 0

Description

The Rajchgot index of a permutation. The '''Rajchgot index''' of a permutation

$\sigma$ is the degree of the ''Grothendieck polynomial'' of

$\sigma$ . This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index [[St000004]] of the permutations smaller than or equal to

$\sigma$ in the right ''weak Bruhat order''.

Matching statistic: St001491

Mapping

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: 8% ●values known / values provided: 19%●distinct values known / distinct values provided: 8%

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 => ? = 6
[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 => ? = 5
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 5
[-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 => ? = 5
[-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 => ? = 5
[-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 => ? = 3
[-1,3,-4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 3
[-1,-3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 3
[-1,-3,-4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 3
[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 => ? = 3
[-1,4,-2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 3
[-1,-4,2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 3
[-1,-4,-2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 3
[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

Mapping

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: 4%

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] => []
 => ?
 => ?
 => ? = 6
[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] => []
 => ?
 => ?
 => ? = 5
[-1,-2,-4,3] => []
 => ?
 => ?
 => ? = 5
[-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

Mapping

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: 4%

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] => []
 => ?
 => ?
 => ? = 6
[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] => []
 => ?
 => ?
 => ? = 5
[-1,-2,-4,3] => []
 => ?
 => ?
 => ? = 5
[-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.

The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

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. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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.