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Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000809: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The reduced reflection length of the permutation.
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth St000029The depth of a permutation. minus the usual length St000018The number of inversions of a permutation..

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

Clarke-Steingrimsson-Zeng

Description

The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where

$des$ is the number of descents, St000021The number of descents of a permutation.,
$exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
$Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
$Ebot$ is the sum of the excedance bottoms,
$Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
$Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
$Res$ is the sum of the (right) embracing numbers,
$Ine$ is the sum of the side numbers.