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Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001171: Permutations ⟶ ℤ (values match St000055The inversion sum of a permutation.)

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6$

$F_{4} = 12 + 12\ q$

Description

The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ .

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

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

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