- FindStat

Identifier

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001605: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.

Map

Demazure product with inverse

Description

This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on signed permutations.
This map is a surjection onto the set of involutions, i.e., the set of signed permutations $\pi$ for which $\pi = \pi^{-1}$.

Map

odd cycle type

Description

The partition corresponding to the odd cycles.
A cycle of length $\ell$ of a signed permutation $\pi$ can be written in two line notation as
$$\begin{array}{cccc} a_1 & a_2 & \dots & a_\ell \\ \pi(a_1) & \pi(a_2) & \dots & \pi(a_\ell) \end{array}$$
where $a_i > 0$ for all $i$, $a_{i+1} = |\pi(a_i)|$ for $i < \ell$ and $a_1 = |\pi(a_\ell)|$.
The cycle is odd, if the number of negative elements in the second row is odd.
This map records the integer partition given by the lengths of the odd cycles.
The integer partition of even cycles together with the integer partition of the odd cycles determines the conjugacy class of the signed permutation.