- FindStat

Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000731: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6$

$F_{4} = 24$

$F_{5} = 110 + 10\ q$

$F_{6} = 660 + 60\ q$

Description

The number of double exceedences of a permutation.
A double exceedence is an index

$\sigma(i)$ such that

$i < \sigma(i) < \sigma(\sigma(i))$ .

Map

Demazure product with inverse

Description

This map sends a permutation

$\pi$ to

$\pi^{-1} \star \pi$ where

$\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations

$\pi$ for which

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

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$