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Your data matches 82 different statistics following compositions of up to 3 maps.
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Matching statistic: St000143

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.

Matching statistic: St000257

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].

Matching statistic: St000929
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => []
 => ? = 0
[1,2] => [1,2] => [1,1]
 => [1]
 => ? = 0
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0
[2,1] => [2,1] => [2]
 => []
 => ? = 0
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0
[-2,1] => [-2,-1] => [2]
 => []
 => ? = 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,2,-3] => [1,2,-3] => [1,1]
 => [1]
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1]
 => []
 => ? = 0
[1,-2,-3] => [1,-2,-3] => [1]
 => []
 => ? = 0
[-1,2,3] => [-1,-2,3] => [1]
 => []
 => ? = 0
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? = 0
[1,3,-2] => [1,-2,3] => [1,1]
 => [1]
 => ? = 0
[1,-3,2] => [1,-3,-2] => [2,1]
 => [1]
 => ? = 0
[1,-3,-2] => [1,-2,-3] => [1]
 => []
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1]
 => []
 => ? = 0
[2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? = 0
[2,1,-3] => [2,1,-3] => [2]
 => []
 => ? = 0
[2,-1,3] => [-1,2,-3] => [1]
 => []
 => ? = 0
[2,-1,-3] => [-1,2,-3] => [1]
 => []
 => ? = 0
[-2,1,3] => [-2,-1,3] => [2,1]
 => [1]
 => ? = 0
[-2,1,-3] => [-2,-1,-3] => [2]
 => []
 => ? = 0
[2,3,1] => [3,2,1] => [2,1]
 => [1]
 => ? = 0
[2,3,-1] => [-1,2,3] => [1,1]
 => [1]
 => ? = 0
[2,-3,1] => [-3,2,-1] => [2,1]
 => [1]
 => ? = 0
[2,-3,-1] => [-1,2,-3] => [1]
 => []
 => ? = 0
[-2,3,1] => [-2,-1,3] => [2,1]
 => [1]
 => ? = 0
[-2,-3,1] => [-2,-1,-3] => [2]
 => []
 => ? = 0
[3,1,2] => [3,2,1] => [2,1]
 => [1]
 => ? = 0
[3,1,-2] => [3,-2,1] => [2]
 => []
 => ? = 0
[3,-1,2] => [-1,-2,3] => [1]
 => []
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1]
 => []
 => ? = 0
[-3,1,2] => [-3,2,-1] => [2,1]
 => [1]
 => ? = 0
[-3,1,-2] => [-3,-2,-1] => [2]
 => []
 => ? = 0
[3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? = 0
[3,2,-1] => [-1,3,2] => [2]
 => []
 => ? = 0
[3,-2,1] => [-2,-1,3] => [2,1]
 => [1]
 => ? = 0
[3,-2,-1] => [-1,-2,3] => [1]
 => []
 => ? = 0
[-3,2,1] => [-3,2,-1] => [2,1]
 => [1]
 => ? = 0
[-3,2,-1] => [-1,-3,-2] => [2]
 => []
 => ? = 0
[-3,-2,1] => [-2,-1,-3] => [2]
 => []
 => ? = 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [1,1]
 => 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => []
 => ? = 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => []
 => ? = 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => []
 => ? = 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
 => []
 => ? = 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
 => [1,1]
 => 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
 => [1,1]
 => 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
 => [1,1]
 => 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
 => [1,1]
 => 1
[1,-3,4,2] => [1,-3,-2,4] => [2,1,1]
 => [1,1]
 => 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,-4,2,3] => [1,-4,3,-2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,-3,2] => [1,-3,-2,4] => [2,1,1]
 => [1,1]
 => 1
[1,-4,3,2] => [1,-4,3,-2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[-2,1,3,4] => [-2,-1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,1,-4,3] => [2,1,-4,-3] => [2,2]
 => [2]
 => 0
[-2,1,4,3] => [-2,-1,4,3] => [2,2]
 => [2]
 => 0
[-2,1,-4,3] => [-2,-1,-4,-3] => [2,2]
 => [2]
 => 0
[2,3,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[2,-3,1,4] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 1
[-2,3,1,4] => [-2,-1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,3,4,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[2,3,4,-1] => [-1,2,3,4] => [1,1,1]
 => [1,1]
 => 1
[2,3,-4,1] => [-4,2,3,-1] => [2,1,1]
 => [1,1]
 => 1
[2,-3,4,1] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 1
[-2,3,4,1] => [-2,-1,4,3] => [2,2]
 => [2]
 => 0
[-2,3,-4,1] => [-2,-1,-4,-3] => [2,2]
 => [2]
 => 0
[2,4,1,3] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[2,-4,1,3] => [-4,2,3,-1] => [2,1,1]
 => [1,1]
 => 1
[-2,4,1,3] => [-2,-1,4,3] => [2,2]
 => [2]
 => 0
[-2,-4,1,3] => [-2,-1,-4,-3] => [2,2]
 => [2]
 => 0
[2,4,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[2,4,-3,1] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 1
[2,-4,3,1] => [-4,2,3,-1] => [2,1,1]
 => [1,1]
 => 1
[-2,4,3,1] => [-2,-1,4,3] => [2,2]
 => [2]
 => 0
[-2,-4,3,1] => [-2,-1,-4,-3] => [2,2]
 => [2]
 => 0
[3,1,2,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[-3,1,2,4] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 1
[3,1,4,2] => [3,4,1,2] => [2,2]
 => [2]
 => 0
[3,1,-4,2] => [3,-4,1,-2] => [2,2]
 => [2]
 => 0
[-3,1,4,2] => [-3,4,-1,2] => [2,2]
 => [2]
 => 0
[-3,1,-4,2] => [-3,-4,-1,-2] => [2,2]
 => [2]
 => 0
[3,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,-2,1,4] => [-2,-1,3,4] => [2,1,1]
 => [1,1]
 => 1
[-3,2,1,4] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 1

Description

The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is

$0$ for partitions

$\lambda \neq 1^n$ and

$1$ for

$\lambda = 1^n$ .

Matching statistic: St001568
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => []
 => ? = 0 + 1
[1,2] => [1,2] => [1,1]
 => [1]
 => ? = 0 + 1
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0 + 1
[2,1] => [2,1] => [2]
 => []
 => ? = 0 + 1
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0 + 1
[-2,1] => [-2,-1] => [2]
 => []
 => ? = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,2,-3] => [1,2,-3] => [1,1]
 => [1]
 => ? = 0 + 1
[1,-2,3] => [1,-2,-3] => [1]
 => []
 => ? = 0 + 1
[1,-2,-3] => [1,-2,-3] => [1]
 => []
 => ? = 0 + 1
[-1,2,3] => [-1,-2,3] => [1]
 => []
 => ? = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? = 0 + 1
[1,3,-2] => [1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => [1]
 => ? = 0 + 1
[1,-3,-2] => [1,-2,-3] => [1]
 => []
 => ? = 0 + 1
[-1,3,2] => [-1,-2,3] => [1]
 => []
 => ? = 0 + 1
[2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? = 0 + 1
[2,1,-3] => [2,1,-3] => [2]
 => []
 => ? = 0 + 1
[2,-1,3] => [-1,2,-3] => [1]
 => []
 => ? = 0 + 1
[2,-1,-3] => [-1,2,-3] => [1]
 => []
 => ? = 0 + 1
[-2,1,3] => [-2,-1,3] => [2,1]
 => [1]
 => ? = 0 + 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => []
 => ? = 0 + 1
[2,3,1] => [3,2,1] => [2,1]
 => [1]
 => ? = 0 + 1
[2,3,-1] => [-1,2,3] => [1,1]
 => [1]
 => ? = 0 + 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => [1]
 => ? = 0 + 1
[2,-3,-1] => [-1,2,-3] => [1]
 => []
 => ? = 0 + 1
[-2,3,1] => [-2,-1,3] => [2,1]
 => [1]
 => ? = 0 + 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => []
 => ? = 0 + 1
[3,1,2] => [3,2,1] => [2,1]
 => [1]
 => ? = 0 + 1
[3,1,-2] => [3,-2,1] => [2]
 => []
 => ? = 0 + 1
[3,-1,2] => [-1,-2,3] => [1]
 => []
 => ? = 0 + 1
[3,-1,-2] => [-1,-2,3] => [1]
 => []
 => ? = 0 + 1
[-3,1,2] => [-3,2,-1] => [2,1]
 => [1]
 => ? = 0 + 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => []
 => ? = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? = 0 + 1
[3,2,-1] => [-1,3,2] => [2]
 => []
 => ? = 0 + 1
[3,-2,1] => [-2,-1,3] => [2,1]
 => [1]
 => ? = 0 + 1
[3,-2,-1] => [-1,-2,3] => [1]
 => []
 => ? = 0 + 1
[-3,2,1] => [-3,2,-1] => [2,1]
 => [1]
 => ? = 0 + 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => []
 => ? = 0 + 1
[-3,-2,1] => [-2,-1,-3] => [2]
 => []
 => ? = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0 + 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0 + 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0 + 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => []
 => ? = 0 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => []
 => ? = 0 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => []
 => ? = 0 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0 + 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
 => []
 => ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0 + 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,-3,4,2] => [1,-3,-2,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,-4,2,3] => [1,-4,3,-2] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,4,-3,2] => [1,-3,-2,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[1,-4,3,2] => [1,-4,3,-2] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[-2,1,3,4] => [-2,-1,3,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 1 = 0 + 1
[2,1,-4,3] => [2,1,-4,-3] => [2,2]
 => [2]
 => 1 = 0 + 1
[-2,1,4,3] => [-2,-1,4,3] => [2,2]
 => [2]
 => 1 = 0 + 1
[-2,1,-4,3] => [-2,-1,-4,-3] => [2,2]
 => [2]
 => 1 = 0 + 1
[2,3,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,-3,1,4] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[-2,3,1,4] => [-2,-1,3,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,3,4,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,3,4,-1] => [-1,2,3,4] => [1,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,3,-4,1] => [-4,2,3,-1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,-3,4,1] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[-2,3,4,1] => [-2,-1,4,3] => [2,2]
 => [2]
 => 1 = 0 + 1
[-2,3,-4,1] => [-2,-1,-4,-3] => [2,2]
 => [2]
 => 1 = 0 + 1
[2,4,1,3] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,-4,1,3] => [-4,2,3,-1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[-2,4,1,3] => [-2,-1,4,3] => [2,2]
 => [2]
 => 1 = 0 + 1
[-2,-4,1,3] => [-2,-1,-4,-3] => [2,2]
 => [2]
 => 1 = 0 + 1
[2,4,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,4,-3,1] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[2,-4,3,1] => [-4,2,3,-1] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[-2,4,3,1] => [-2,-1,4,3] => [2,2]
 => [2]
 => 1 = 0 + 1
[-2,-4,3,1] => [-2,-1,-4,-3] => [2,2]
 => [2]
 => 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[-3,1,2,4] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,1,4,2] => [3,4,1,2] => [2,2]
 => [2]
 => 1 = 0 + 1
[3,1,-4,2] => [3,-4,1,-2] => [2,2]
 => [2]
 => 1 = 0 + 1
[-3,1,4,2] => [-3,4,-1,2] => [2,2]
 => [2]
 => 1 = 0 + 1
[-3,1,-4,2] => [-3,-4,-1,-2] => [2,2]
 => [2]
 => 1 = 0 + 1
[3,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[3,-2,1,4] => [-2,-1,3,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1
[-3,2,1,4] => [-3,2,-1,4] => [2,1,1]
 => [1,1]
 => 2 = 1 + 1

Description

The smallest positive integer that does not appear twice in the partition.

Matching statistic: St001604
(load all 6 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => []
 => ? = 0
[1,2] => [1,2] => []
 => ? = 0
[1,-2] => [1,-2] => [1]
 => ? = 0
[2,1] => [2,1] => []
 => ? = 0
[2,-1] => [-1,2] => [1]
 => ? = 0
[-2,1] => [-2,-1] => []
 => ? = 0
[1,2,3] => [1,2,3] => []
 => ? = 1
[1,2,-3] => [1,2,-3] => [1]
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1,1]
 => ? = 0
[1,-2,-3] => [1,-2,-3] => [1,1]
 => ? = 0
[-1,2,3] => [-1,-2,3] => [1,1]
 => ? = 0
[1,3,2] => [1,3,2] => []
 => ? = 0
[1,3,-2] => [1,-2,3] => [1]
 => ? = 0
[1,-3,2] => [1,-3,-2] => []
 => ? = 0
[1,-3,-2] => [1,-2,-3] => [1,1]
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1,1]
 => ? = 0
[2,1,3] => [2,1,3] => []
 => ? = 0
[2,1,-3] => [2,1,-3] => [1]
 => ? = 0
[2,-1,3] => [-1,2,-3] => [1,1]
 => ? = 0
[2,-1,-3] => [-1,2,-3] => [1,1]
 => ? = 0
[-2,1,3] => [-2,-1,3] => []
 => ? = 0
[-2,1,-3] => [-2,-1,-3] => [1]
 => ? = 0
[2,3,1] => [3,2,1] => []
 => ? = 0
[2,3,-1] => [-1,2,3] => [1]
 => ? = 0
[2,-3,1] => [-3,2,-1] => []
 => ? = 0
[2,-3,-1] => [-1,2,-3] => [1,1]
 => ? = 0
[-2,3,1] => [-2,-1,3] => []
 => ? = 0
[-2,-3,1] => [-2,-1,-3] => [1]
 => ? = 0
[3,1,2] => [3,2,1] => []
 => ? = 0
[3,1,-2] => [3,-2,1] => [1]
 => ? = 0
[3,-1,2] => [-1,-2,3] => [1,1]
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1,1]
 => ? = 0
[-3,1,2] => [-3,2,-1] => []
 => ? = 0
[-3,1,-2] => [-3,-2,-1] => [1]
 => ? = 0
[3,2,1] => [3,2,1] => []
 => ? = 0
[3,2,-1] => [-1,3,2] => [1]
 => ? = 0
[3,-2,1] => [-2,-1,3] => []
 => ? = 0
[3,-2,-1] => [-1,-2,3] => [1,1]
 => ? = 0
[-3,2,1] => [-3,2,-1] => []
 => ? = 0
[-3,2,-1] => [-1,-3,-2] => [1]
 => ? = 0
[-3,-2,1] => [-2,-1,-3] => [1]
 => ? = 0
[1,2,3,4] => [1,2,3,4] => []
 => ? = 1
[1,2,3,-4] => [1,2,3,-4] => [1]
 => ? = 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => ? = 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => ? = 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => ? = 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => ? = 0
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => []
 => ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1]
 => ? = 1
[1,2,-4,3] => [1,2,-4,-3] => []
 => ? = 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
 => 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
 => 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
 => 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
 => 0
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
 => 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
 => 0
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => 0
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
 => 0
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
 => 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
 => 0
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
 => 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1,1,1]
 => 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => 0
[1,-4,-3,-2] => [1,-2,-3,-4] => [1,1,1]
 => 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1,1,1]
 => 0
[2,-1,3,-4] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-1,-3,4] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-1,-3,-4] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-1,4,-3] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-1,-4,3] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-1,-4,-3] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-3,-1,4] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-3,-1,-4] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-3,4,-1] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-3,-4,-1] => [-1,2,-3,-4] => [1,1,1]
 => 0
[-2,3,4,-1] => [-1,-2,-3,4] => [1,1,1]
 => 0
[2,-4,-1,3] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-4,-1,-3] => [-1,2,-3,-4] => [1,1,1]
 => 0
[2,-4,-3,-1] => [-1,2,-3,-4] => [1,1,1]
 => 0
[-2,4,3,-1] => [-1,-2,-3,4] => [1,1,1]
 => 0
[3,-1,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-1,-2,4] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-1,-2,-4] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-1,4,-2] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-1,-4,2] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-1,-4,-2] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-2,-1,4] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-2,-1,-4] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-2,4,-1] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-2,-4,-1] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-4,-1,2] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-4,-1,-2] => [-1,-2,3,-4] => [1,1,1]
 => 0
[3,-4,-2,-1] => [-1,-2,3,-4] => [1,1,1]
 => 0
[4,-1,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => 0
[4,-1,-2,3] => [-1,-2,-3,4] => [1,1,1]
 => 0
[4,-1,-2,-3] => [-1,-2,-3,4] => [1,1,1]
 => 0

Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St000478

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => []
 => ?
 => ? = 0
[1,2] => [1,2] => []
 => ?
 => ? = 0
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0
[2,1] => [2,1] => []
 => ?
 => ? = 0
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 1
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 0
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 0
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 0
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 0
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 0
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 0
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 0
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 0
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 0
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 0
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 0
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 1
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => []
 => ?
 => ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1]
 => []
 => ? = 1
[1,2,-4,3] => [1,2,-4,-3] => []
 => ?
 => ? = 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-3,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,3,4,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-3,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,4,3,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-2,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0

Description

Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1]

$\sum_{\pi\in GG_1(r)} w_1(\pi)$ equals the number of partitions of

$r$ whose odd parts are all distinct.

$GG_1(r)$ is the set of partitions of

$r$ where consecutive entries differ by at least

$2$ , and consecutive even entries differ by at least

$4$ .

Matching statistic: St000566

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => []
 => ?
 => ? = 0
[1,2] => [1,2] => []
 => ?
 => ? = 0
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0
[2,1] => [2,1] => []
 => ?
 => ? = 0
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 1
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 0
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 0
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 0
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 0
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 0
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 0
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 0
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 0
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 0
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 0
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 0
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 1
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => []
 => ?
 => ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1]
 => []
 => ? = 1
[1,2,-4,3] => [1,2,-4,-3] => []
 => ?
 => ? = 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-3,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,3,4,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-3,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,4,3,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-2,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0

Description

The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if

$\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is

$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$

Matching statistic: St000621

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => []
 => ?
 => ? = 0
[1,2] => [1,2] => []
 => ?
 => ? = 0
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0
[2,1] => [2,1] => []
 => ?
 => ? = 0
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 1
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 0
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 0
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 0
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 0
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 0
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 0
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 0
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 0
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 0
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 0
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 0
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 1
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => []
 => ?
 => ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1]
 => []
 => ? = 1
[1,2,-4,3] => [1,2,-4,-3] => []
 => ?
 => ? = 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-3,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,3,4,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-3,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,4,3,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-2,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0

Description

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. To be precise, this is given for a partition

$\lambda \vdash n$ by the number of standard tableaux

$T$ of shape

$\lambda$ such that

$\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]].

Matching statistic: St000934

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => []
 => ?
 => ? = 0
[1,2] => [1,2] => []
 => ?
 => ? = 0
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0
[2,1] => [2,1] => []
 => ?
 => ? = 0
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 1
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 0
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 0
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 0
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 0
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 0
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 0
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 0
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 0
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 0
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 0
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 0
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 1
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => []
 => ?
 => ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1]
 => []
 => ? = 1
[1,2,-4,3] => [1,2,-4,-3] => []
 => ?
 => ? = 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-3,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,3,4,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-3,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,4,3,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-2,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0

Description

The 2-degree of an integer partition. For an integer partition

$\lambda$ , this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by

$\lambda$ .

Matching statistic: St000936

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => []
 => ?
 => ? = 0
[1,2] => [1,2] => []
 => ?
 => ? = 0
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0
[2,1] => [2,1] => []
 => ?
 => ? = 0
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 1
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 0
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 0
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 0
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 0
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 0
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 0
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 0
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 0
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 0
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 0
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 0
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 0
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 1
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0
[-1,2,3,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => []
 => ?
 => ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1]
 => []
 => ? = 1
[1,2,-4,3] => [1,2,-4,-3] => []
 => ?
 => ? = 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-3,-4,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-4,-3,-2] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-3,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-1,-4,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-1,-4] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-3,-4,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,3,4,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-1,-3] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[2,-4,-3,-1] => [-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 0
[-2,4,3,-1] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-2,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-1,-4,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-1,-4] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-2,-4,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-1,-2] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[3,-4,-2,-1] => [-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,3] => [-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[4,-1,-2,-3] => [-1,-2,-3,4] => [1,1,1]
 => [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 72 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. 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. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001964The interval resolution global dimension of a poset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.