- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000040: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

[1,1,1,0,0,0] => [3,2,1] => 6

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

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

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

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

[1,0,1,1,1,0,0,0] => [1,4,3,2] => 6

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

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

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

[1,1,0,1,0,1,0,0] => [2,3,4,1] => 8

[1,1,0,1,1,0,0,0] => [2,4,3,1] => 12

[1,1,1,0,0,0,1,0] => [3,2,1,4] => 6

[1,1,1,0,0,1,0,0] => [3,2,4,1] => 12

[1,1,1,0,1,0,0,0] => [3,4,2,1] => 18

[1,1,1,1,0,0,0,0] => [4,3,2,1] => 24

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

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

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

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

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 6

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

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

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

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 8

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 12

[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 6

[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 12

[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 18

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 24

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

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

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

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 8

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 12

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

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 8

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 8

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 16

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 24

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 12

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 24

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 36

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 48

[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 6

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 12

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 12

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 24

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 36

[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 18

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 36

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 54

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 72

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 24

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 48

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 72

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 96

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 120

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 2

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 2

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 4

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 6

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 2

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 4

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 4

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 8

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 12

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 6

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 12

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 18

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 24

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 2

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 4

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 4

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 8

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 12

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 4

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 8

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 8

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 16

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 24

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 12

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 24

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 36

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 48

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 6

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 12

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 12

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 24

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 36

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => 18

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 36

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 54

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 72

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + 2\ q^{2} + q^{4} + q^{6}$

$F_{4} = q + 3\ q^{2} + 3\ q^{4} + 2\ q^{6} + q^{8} + 2\ q^{12} + q^{18} + q^{24}$

$F_{5} = q + 4\ q^{2} + 6\ q^{4} + 3\ q^{6} + 4\ q^{8} + 6\ q^{12} + q^{16} + 2\ q^{18} + 5\ q^{24} + 3\ q^{36} + 2\ q^{48} + q^{54} + 2\ q^{72} + q^{96} + q^{120}$

$F_{6} = q + 5\ q^{2} + 10\ q^{4} + 4\ q^{6} + 10\ q^{8} + 12\ q^{12} + 5\ q^{16} + 3\ q^{18} + 15\ q^{24} + q^{32} + 9\ q^{36} + 10\ q^{48} + 2\ q^{54} + 10\ q^{72} + 5\ q^{96} + 4\ q^{108} + 2\ q^{120} + 6\ q^{144} + q^{162} + 2\ q^{192} + 3\ q^{216} + 2\ q^{240} + 3\ q^{288} + 2\ q^{360} + q^{384} + 2\ q^{480} + q^{600} + q^{720}$

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

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).