- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
St000040: Permutations ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,3,4,5,2] => [1,3,4,5,2] => 8

[1,3,5,2,4] => [1,3,5,2,4] => 8

[1,3,5,4,2] => [1,3,5,2,4] => 8

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

[1,4,2,5,3] => [1,4,2,5,3] => 8

[1,4,3,2,5] => [1,4,2,5,3] => 8

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

[1,4,5,2,3] => [1,4,5,2,3] => 14

[1,4,5,3,2] => [1,4,5,2,3] => 14

[1,5,2,3,4] => [1,5,2,3,4] => 8

[1,5,2,4,3] => [1,5,2,4,3] => 12

[1,5,3,2,4] => [1,5,2,4,3] => 12

[1,5,3,4,2] => [1,5,2,3,4] => 8

[1,5,4,2,3] => [1,5,2,3,4] => 8

[1,5,4,3,2] => [1,5,2,3,4] => 8

[2,1,3,4,5] => [1,3,4,5,2] => 8

[2,1,3,5,4] => [1,3,5,2,4] => 8

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

[2,1,4,5,3] => [1,4,5,2,3] => 14

[2,1,5,3,4] => [1,5,2,3,4] => 8

[2,1,5,4,3] => [1,5,2,3,4] => 8

[2,3,1,4,5] => [1,4,5,2,3] => 14

[2,3,1,5,4] => [1,5,2,3,4] => 8

[2,3,4,1,5] => [1,5,2,3,4] => 8

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

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

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

[2,4,1,3,5] => [1,3,5,2,4] => 8

[2,4,1,5,3] => [1,5,2,4,3] => 12

[2,4,3,1,5] => [1,5,2,4,3] => 12

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

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

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

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

[2,5,1,4,3] => [1,4,2,5,3] => 8

[2,5,3,1,4] => [1,4,2,5,3] => 8

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

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

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

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

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

[3,1,4,2,5] => [1,4,2,5,3] => 8

[3,1,4,5,2] => [1,4,5,2,3] => 14

[3,1,5,2,4] => [1,5,2,4,3] => 12

[3,1,5,4,2] => [1,5,2,3,4] => 8

[3,2,1,4,5] => [1,4,5,2,3] => 14

[3,2,1,5,4] => [1,5,2,3,4] => 8

[3,2,4,1,5] => [1,5,2,4,3] => 12

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

[3,2,5,1,4] => [1,4,2,5,3] => 8

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

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

[3,4,1,5,2] => [1,5,2,3,4] => 8

[3,4,2,1,5] => [1,5,2,3,4] => 8

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

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

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

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

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

>>> Load all 4725 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of regions of the inversion arrangement of a permutation.
The inversion arrangement $\mathcal{A}_w$ consists of the hyperplanes $x_i-x_j=0$ such that $(i,j)$ is an inversion of $w$.
Postnikov [4] conjectured that the number of regions in $\mathcal{A}_w$ equals the number of permutations in the interval $[id,w]$ in the strong Bruhat order if and only if $w$ avoids $4231$, $35142$, $42513$, $351624$. This conjecture was proved by Hultman-Linusson-Shareshian-Sjöstrand [1].
Oh-Postnikov-Yoo [3] showed that the number of regions of $\mathcal{A}_w$ is $|\chi_{G_w}(-1)|$ where $\chi_{G_w}$ is the chromatic polynomial of the inversion graph $G_w$. This is the graph with vertices ${1,2,\ldots,n}$ and edges $(i,j)$ for $i\lneq j$ $w_i\gneq w_j$.
For a permutation $w=w_1\cdots w_n$, Lewis-Morales [2] and Hultman (see appendix in [2]) showed that this number equals the number of placements of $n$ non-attacking rooks on the south-west Rothe diagram of $w$.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.