- FindStat

Identifier

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 ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1299 entries. <<<

search for individual values

searching the database for the individual values of this statistic

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.

Map

first row removal

Description

Removes the first entry of an integer partition

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.

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

$\pi = \pi^{-1}$ .