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Matching statistic: St000506

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000506: 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
[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]
 => 0
[1,2,-3] => [1,2,-3] => [1,1]
 => 1
[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
[1,3,-2] => [1,-2,3] => [1,1]
 => 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => 1
[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
[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
[-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]
 => 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => 1
[2,-3,-1] => [-1,2,-3] => [1]
 => 0
[-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]
 => 0
[3,-1,-2] => [-1,-2,3] => [1]
 => 0
[-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]
 => 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]
 => 0
[1,2,3,4] => [1,2,3,4] => [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
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => 1
[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

Description

The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry

$i$ such that

$i+1$ appears to the right or above

$i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.

Matching statistic: St000621

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => [1]
 => ? = 0
[1,2] => [1,2] => [1,1]
 => [2]
 => 1
[1,-2] => [1,-2] => [1]
 => [1]
 => ? = 0
[2,1] => [2,1] => [2]
 => [1,1]
 => 0
[2,-1] => [-1,2] => [1]
 => [1]
 => ? = 0
[-2,1] => [-2,-1] => [2]
 => [1,1]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 0
[1,2,-3] => [1,2,-3] => [1,1]
 => [2]
 => 1
[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] => [2,1]
 => [2,1]
 => 1
[1,3,-2] => [1,-2,3] => [1,1]
 => [2]
 => 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => [2,1]
 => 1
[1,-3,-2] => [1,-2,-3] => [1]
 => [1]
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1]
 => [1]
 => ? = 0
[2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1
[2,1,-3] => [2,1,-3] => [2]
 => [1,1]
 => 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] => [2,1]
 => [2,1]
 => 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => [1,1]
 => 0
[2,3,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1
[2,3,-1] => [-1,2,3] => [1,1]
 => [2]
 => 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => [2,1]
 => 1
[2,-3,-1] => [-1,2,-3] => [1]
 => [1]
 => ? = 0
[-2,3,1] => [-2,-1,3] => [2,1]
 => [2,1]
 => 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => [1,1]
 => 0
[3,1,2] => [3,2,1] => [2,1]
 => [2,1]
 => 1
[3,1,-2] => [3,-2,1] => [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] => [2,1]
 => [2,1]
 => 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => [1,1]
 => 0
[3,2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1
[3,2,-1] => [-1,3,2] => [2]
 => [1,1]
 => 0
[3,-2,1] => [-2,-1,3] => [2,1]
 => [2,1]
 => 1
[3,-2,-1] => [-1,-2,3] => [1]
 => [1]
 => ? = 0
[-3,2,1] => [-3,2,-1] => [2,1]
 => [2,1]
 => 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => [1,1]
 => 0
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,1]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [3]
 => 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [2]
 => 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [2]
 => 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [2]
 => 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[1,-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]
 => [2]
 => 1
[-1,2,3,-4] => [-1,-2,3,-4] => [1]
 => [1]
 => ? = 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [3,1]
 => 1
[1,2,4,-3] => [1,2,-3,4] => [1,1,1]
 => [3]
 => 0
[1,2,-4,3] => [1,2,-4,-3] => [2,1,1]
 => [3,1]
 => 1
[1,2,-4,-3] => [1,2,-3,-4] => [1,1]
 => [2]
 => 1
[1,-2,4,3] => [1,-2,-3,4] => [1,1]
 => [2]
 => 1
[1,-2,4,-3] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[1,-2,-4,3] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[1,-2,-4,-3] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[-1,2,4,3] => [-1,-2,4,3] => [2]
 => [1,1]
 => 0
[-1,2,4,-3] => [-1,-2,-3,4] => [1]
 => [1]
 => ? = 0
[-1,2,-4,3] => [-1,-2,-4,-3] => [2]
 => [1,1]
 => 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [3,1]
 => 1
[1,3,2,-4] => [1,3,2,-4] => [2,1]
 => [2,1]
 => 1
[1,3,-2,4] => [1,-2,3,-4] => [1,1]
 => [2]
 => 1
[1,3,-2,-4] => [1,-2,3,-4] => [1,1]
 => [2]
 => 1
[1,-3,2,4] => [1,-3,-2,4] => [2,1,1]
 => [3,1]
 => 1
[1,-3,2,-4] => [1,-3,-2,-4] => [2,1]
 => [2,1]
 => 1
[1,-3,-2,4] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[1,-3,-2,-4] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[-1,3,2,4] => [-1,-2,3,4] => [1,1]
 => [2]
 => 1
[-1,3,2,-4] => [-1,-2,3,-4] => [1]
 => [1]
 => ? = 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
 => [3,1]
 => 1
[1,3,4,-2] => [1,-2,3,4] => [1,1,1]
 => [3]
 => 0
[1,3,-4,2] => [1,-4,3,-2] => [2,1,1]
 => [3,1]
 => 1
[1,-3,4,-2] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[1,-3,-4,-2] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[-1,3,4,-2] => [-1,-2,-3,4] => [1]
 => [1]
 => ? = 0
[1,-4,-2,3] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[1,-4,-2,-3] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[-1,4,2,-3] => [-1,-2,-3,4] => [1]
 => [1]
 => ? = 0
[1,-4,-3,-2] => [1,-2,-3,-4] => [1]
 => [1]
 => ? = 0
[-1,4,3,-2] => [-1,-2,-3,4] => [1]
 => [1]
 => ? = 0
[2,-1,3,-4] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-1,-3,4] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-1,-3,-4] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-1,4,-3] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-1,-4,3] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-1,-4,-3] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-3,-1,4] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-3,-1,-4] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-3,4,-1] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-3,-4,-1] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[-2,3,4,-1] => [-1,-2,-3,4] => [1]
 => [1]
 => ? = 0
[2,-4,-1,3] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-4,-1,-3] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[2,-4,-3,-1] => [-1,2,-3,-4] => [1]
 => [1]
 => ? = 0
[-2,4,3,-1] => [-1,-2,-3,4] => [1]
 => [1]
 => ? = 0
[3,-1,2,-4] => [-1,-2,3,-4] => [1]
 => [1]
 => ? = 0
[3,-1,-2,4] => [-1,-2,3,-4] => [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: 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: 33%

Values

[1] => [1] => []
 => ?
 => ? = 0
[1,2] => [1,2] => []
 => ?
 => ? = 1
[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] => []
 => ?
 => ? = 0
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 1
[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] => []
 => ?
 => ? = 1
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 1
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 1
[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] => []
 => ?
 => ? = 1
[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] => []
 => ?
 => ? = 1
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 0
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 1
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 1
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 1
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-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]
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-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]
 => ? = 0
[-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] => []
 => ?
 => ? = 1
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 1
[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]
 => ? = 1
[1,-2,3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,3,-4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,-3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,3,-5,-4] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,-3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,4,-3,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,4,-3,-5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,4,3,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,-5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,4,-5,-3] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,4,5,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,5,-3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,5,-3,-4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,5,3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,5,-4,-3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,5,4,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-3,-2,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [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: 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: 33%

Values

[1] => [1] => []
 => ?
 => ? = 0 + 1
[1,2] => [1,2] => []
 => ?
 => ? = 1 + 1
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0 + 1
[2,1] => [2,1] => []
 => ?
 => ? = 0 + 1
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0 + 1
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0 + 1
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 0 + 1
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 1 + 1
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0 + 1
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0 + 1
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 1
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 1 + 1
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 1 + 1
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 1 + 1
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0 + 1
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 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]
 => ? = 0 + 1
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0 + 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]
 => []
 => ? = 1 + 1
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 1 + 1
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0 + 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]
 => ? = 0 + 1
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 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]
 => ? = 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]
 => []
 => ? = 0 + 1
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 1 + 1
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 0 + 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 1 + 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 1 + 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 1 + 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 1 + 1
[1,-2,3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,3,-4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,-3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,3,-5,-4] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,-3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,4,-3,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,4,-3,-5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,4,3,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,-5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,4,-5,-3] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,4,5,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,5,-3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,5,-3,-4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,5,3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,5,-4,-3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,5,4,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-3,-2,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 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: 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: 33%

Values

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

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.