- FindStat

Identifier

Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions

Images

[1] => [-1] => []

[-1] => [1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

[-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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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,-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]

[-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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 4282 entries. <<<

Map

bar

Description

Return the signed permutation with all signs reversed.

Map

even cycle type

Description

The partition corresponding to the even 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 even, if the number of negative elements in the second row is even.
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.