- FindStat

Identifier

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001440: Integer partitions ⟶ ℤ

Values

[-1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of standard Young tableaux whose major index is congruent one modulo the size of a given 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.