- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ 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] => [1,2,3] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,1,2,3] => [1,4,3,2] => 6

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

[4,2,1,3] => [1,4,3,2] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,5,2,3,4] => [1,2,5,4,3] => 6

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

[1,5,3,2,4] => [1,2,5,4,3] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,5,1,3,4] => [1,2,5,4,3] => 6

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

[2,5,3,1,4] => [1,2,5,4,3] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 3911 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q + 2\ q^{2}$

$F_{4} = 8\ q + 8\ q^{2} + 6\ q^{4} + 2\ q^{6}$

$F_{5} = 16\ q + 24\ q^{2} + 28\ q^{4} + 8\ q^{6} + 18\ q^{8} + 12\ q^{12} + 4\ q^{14} + 8\ q^{18} + 2\ q^{24}$

$F_{6} = 32\ q + 64\ q^{2} + 96\ q^{4} + 24\ q^{6} + 96\ q^{8} + 56\ q^{12} + 16\ q^{14} + 54\ q^{16} + 32\ q^{18} + 62\ q^{24} + 24\ q^{28} + 54\ q^{36} + 8\ q^{42} + 12\ q^{46} + 12\ q^{48} + 24\ q^{54} + 16\ q^{60} + 16\ q^{72} + 10\ q^{78} + 10\ q^{96} + 2\ q^{120}$

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

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.