Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001939
Mapping
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => 1
[1,2] => [1,2] => [1,1]
 => 0
[1,-2] => [1,-2] => [1]
 => 1
[2,1] => [2,1] => [2]
 => 0
[2,-1] => [-1,2] => [1]
 => 1
[-2,1] => [-2,-1] => [2]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => 0
[1,2,-3] => [1,2,-3] => [1,1]
 => 0
[1,-2,3] => [1,-2,-3] => [1]
 => 1
[1,-2,-3] => [1,-2,-3] => [1]
 => 1
[-1,2,3] => [-1,-2,3] => [1]
 => 1
[1,3,2] => [1,3,2] => [2,1]
 => 1
[1,3,-2] => [1,-2,3] => [1,1]
 => 0
[1,-3,2] => [1,-3,-2] => [2,1]
 => 1
[1,-3,-2] => [1,-2,-3] => [1]
 => 1
[-1,3,2] => [-1,-2,3] => [1]
 => 1
[2,1,3] => [2,1,3] => [2,1]
 => 1
[2,1,-3] => [2,1,-3] => [2]
 => 0
[2,-1,3] => [-1,2,-3] => [1]
 => 1
[2,-1,-3] => [-1,2,-3] => [1]
 => 1
[-2,1,3] => [-2,-1,3] => [2,1]
 => 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => 0
[2,3,1] => [3,2,1] => [2,1]
 => 1
[2,3,-1] => [-1,2,3] => [1,1]
 => 0
[2,-3,1] => [-3,2,-1] => [2,1]
 => 1
[2,-3,-1] => [-1,2,-3] => [1]
 => 1
[-2,3,1] => [-2,-1,3] => [2,1]
 => 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => 0
[3,1,2] => [3,2,1] => [2,1]
 => 1
[3,1,-2] => [3,-2,1] => [2]
 => 0
[3,-1,2] => [-1,-2,3] => [1]
 => 1
[3,-1,-2] => [-1,-2,3] => [1]
 => 1
[-3,1,2] => [-3,2,-1] => [2,1]
 => 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => 0
[3,2,1] => [3,2,1] => [2,1]
 => 1
[3,2,-1] => [-1,3,2] => [2]
 => 0
[3,-2,1] => [-2,-1,3] => [2,1]
 => 1
[3,-2,-1] => [-1,-2,3] => [1]
 => 1
[-3,2,1] => [-3,2,-1] => [2,1]
 => 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => 0
[-3,-2,1] => [-2,-1,-3] => [2]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 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]
 => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => 0
Description
The number of parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001603
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
St001603: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => []
 => ?
 => ? = 1
[1,2] => [1,2] => []
 => ?
 => ? = 0
[1,-2] => [1,-2] => [1]
 => []
 => ? = 1
[2,1] => [2,1] => []
 => ?
 => ? = 0
[2,-1] => [-1,2] => [1]
 => []
 => ? = 1
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 0
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 1
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 1
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 1
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 1
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 1
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 1
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 0
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 1
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 1
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 1
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 1
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 1
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 1
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 1
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 1
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 0
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 1
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 0
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 1
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 0
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 1
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 1
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 0
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 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] => [1,1]
 => [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]
 => ? = 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,3,-4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,-3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,-2,3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-3,5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,3,-5,-4] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,-3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,-2,3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-4,3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-4,3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,4,-3,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,4,-3,-5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,-2,4,3,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-4,5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-4,5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-4,-5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,4,-5,-3] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,-2,4,5,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-5,3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,5,-3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,5,-3,-4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,-2,5,3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-5,4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-2,-5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,2,5,-4,-3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[-1,-2,5,4,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,-3,-2,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001604
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
St001604: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => []
 => ?
 => ? = 1 - 1
[1,2] => [1,2] => []
 => ?
 => ? = 0 - 1
[1,-2] => [1,-2] => [1]
 => []
 => ? = 1 - 1
[2,1] => [2,1] => []
 => ?
 => ? = 0 - 1
[2,-1] => [-1,2] => [1]
 => []
 => ? = 1 - 1
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0 - 1
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 0 - 1
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0 - 1
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 1 - 1
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 1 - 1
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 - 1
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 1 - 1
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0 - 1
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 1 - 1
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 1 - 1
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 - 1
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 1 - 1
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 0 - 1
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 1 - 1
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 1 - 1
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 1 - 1
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 0 - 1
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 1 - 1
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0 - 1
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 1 - 1
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 1 - 1
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 1 - 1
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0 - 1
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 1 - 1
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 0 - 1
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 - 1
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 - 1
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 1 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 0 - 1
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 1 - 1
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 0 - 1
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 1 - 1
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 - 1
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 1 - 1
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 0 - 1
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0 - 1
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 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] => [1,1]
 => [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]
 => [1,1]
 => ? = 1 - 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 1 - 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 1 - 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0 - 1
[1,-2,3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,3,-4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,-3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,-2,3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-3,5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,3,-5,-4] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,-3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,-2,3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-4,3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-4,3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,4,-3,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,4,-3,-5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,-2,4,3,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-4,5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-4,5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-4,-5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,4,-5,-3] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,-2,4,5,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-5,3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,5,-3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,5,-3,-4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,-2,5,3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-5,4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-2,-5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,2,5,-4,-3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-1,-2,5,4,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,-3,-2,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
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: St001605
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
St001605: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => []
 => ?
 => ? = 1 + 1
[1,2] => [1,2] => []
 => ?
 => ? = 0 + 1
[1,-2] => [1,-2] => [1]
 => []
 => ? = 1 + 1
[2,1] => [2,1] => []
 => ?
 => ? = 0 + 1
[2,-1] => [-1,2] => [1]
 => []
 => ? = 1 + 1
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0 + 1
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 0 + 1
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0 + 1
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 1 + 1
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 1 + 1
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 + 1
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 1 + 1
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0 + 1
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 1 + 1
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 1 + 1
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 + 1
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 1 + 1
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 0 + 1
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 1 + 1
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 1 + 1
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 1 + 1
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 0 + 1
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 1 + 1
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0 + 1
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 1 + 1
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 1 + 1
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 1 + 1
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0 + 1
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 1 + 1
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 0 + 1
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 + 1
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 + 1
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 1 + 1
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 0 + 1
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 1 + 1
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 0 + 1
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 1 + 1
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 1 + 1
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 1 + 1
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 0 + 1
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 0 + 1
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 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] => [1,1]
 => [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]
 => [1,1]
 => ? = 1 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 1 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 1 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0 + 1
[1,-2,3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,3,-4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,-3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,-2,3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-3,5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,3,-5,-4] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,-3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,-2,3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-4,3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-4,3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,4,-3,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,4,-3,-5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,-2,4,3,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-4,5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-4,5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-4,-5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,4,-5,-3] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,-2,4,5,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-5,3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,5,-3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,5,-3,-4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,-2,5,3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-5,4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-2,-5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,2,5,-4,-3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[-1,-2,5,4,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
[1,-3,-2,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 1 + 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.