- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000365: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6$

$F_{4} = 24$

$F_{5} = 118 + 2\ q$

$F_{6} = 702 + 18\ q$

Description

The number of double ascents of a permutation.
A double ascent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i) < \pi(i+1) < \pi(i+2)$ .

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

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].