- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000040: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2427 entries. <<<

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$F_{(2, 2)} = 2\ q + q^{2}$

$F_{(2, 3)} = 3\ q + 2\ q^{2}$

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

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

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

$F_{(4, 2)} = 4\ q + q^{2} + q^{4} + q^{8} + q^{14}$

$F_{(2, 5)} = 5\ q + 4\ q^{2}$

$F_{(3, 4)} = 10\ q + 6\ q^{2} + 6\ q^{4} + 3\ q^{6}$

$F_{(4, 3)} = 10\ q + 4\ q^{2} + 4\ q^{4} + q^{6} + 5\ q^{8} + q^{12} + 4\ q^{14} + q^{18}$

$F_{(5, 2)} = 5\ q + q^{2} + q^{4} + q^{8} + q^{14} + q^{16} + q^{46}$

$F_{(2, 6)} = 6\ q + 5\ q^{2}$

$F_{(3, 5)} = 15\ q + 10\ q^{2} + 10\ q^{4} + 6\ q^{6}$

$F_{(4, 4)} = 20\ q + 10\ q^{2} + 10\ q^{4} + 4\ q^{6} + 14\ q^{8} + 4\ q^{12} + 10\ q^{14} + 4\ q^{18} + q^{24}$

$F_{(5, 3)} = 15\ q + 5\ q^{2} + 5\ q^{4} + q^{6} + 6\ q^{8} + q^{12} + 5\ q^{14} + 6\ q^{16} + q^{18} + q^{24} + q^{28} + q^{36} + q^{42} + 5\ q^{46} + q^{54} + q^{60} + q^{78}$

$F_{(6, 2)} = 6\ q + q^{2} + q^{4} + q^{8} + q^{14} + q^{16} + q^{32} + q^{46} + q^{146} + q^{230}$

$F_{(2, 7)} = 7\ q + 6\ q^{2}$

$F_{(3, 6)} = 21\ q + 15\ q^{2} + 15\ q^{4} + 10\ q^{6}$

$F_{(4, 5)} = 35\ q + 20\ q^{2} + 20\ q^{4} + 10\ q^{6} + 30\ q^{8} + 10\ q^{12} + 20\ q^{14} + 10\ q^{18} + 4\ q^{24}$

$F_{(5, 4)} = 35\ q + 15\ q^{2} + 15\ q^{4} + 5\ q^{6} + 20\ q^{8} + 5\ q^{12} + 15\ q^{14} + 20\ q^{16} + 5\ q^{18} + 7\ q^{24} + 5\ q^{28} + 5\ q^{36} + 5\ q^{42} + 15\ q^{46} + q^{48} + 6\ q^{54} + 5\ q^{60} + q^{72} + 5\ q^{78} + q^{96}$

$F_{(6, 3)} = 21\ q + 6\ q^{2} + 6\ q^{4} + q^{6} + 7\ q^{8} + q^{12} + 6\ q^{14} + 7\ q^{16} + q^{18} + q^{24} + q^{28} + 7\ q^{32} + q^{36} + q^{42} + 6\ q^{46} + q^{48} + q^{54} + q^{56} + q^{60} + q^{72} + q^{78} + 3\ q^{92} + q^{108} + q^{138} + 6\ q^{146} + q^{152} + q^{156} + q^{162} + q^{198} + 6\ q^{230} + q^{276} + q^{330} + q^{426}$

$F_{(7, 2)} = 7\ q + q^{2} + q^{4} + q^{8} + q^{14} + q^{16} + q^{32} + q^{46} + q^{64} + q^{146} + q^{230} + q^{454} + q^{1066}$

$F_{(2, 8)} = 8\ q + 7\ q^{2}$

$F_{(3, 7)} = 28\ q + 21\ q^{2} + 21\ q^{4} + 15\ q^{6}$

$F_{(4, 6)} = 56\ q + 35\ q^{2} + 35\ q^{4} + 20\ q^{6} + 55\ q^{8} + 20\ q^{12} + 35\ q^{14} + 20\ q^{18} + 10\ q^{24}$

$F_{(5, 5)} = 70\ q + 35\ q^{2} + 35\ q^{4} + 15\ q^{6} + 50\ q^{8} + 15\ q^{12} + 35\ q^{14} + 50\ q^{16} + 15\ q^{18} + 25\ q^{24} + 15\ q^{28} + 15\ q^{36} + 15\ q^{42} + 35\ q^{46} + 5\ q^{48} + 20\ q^{54} + 15\ q^{60} + 5\ q^{72} + 15\ q^{78} + 5\ q^{96} + q^{120}$

$F_{(6, 4)} = 56\ q + 21\ q^{2} + 21\ q^{4} + 6\ q^{6} + 27\ q^{8} + 6\ q^{12} + 21\ q^{14} + 27\ q^{16} + 6\ q^{18} + 8\ q^{24} + 6\ q^{28} + 27\ q^{32} + 6\ q^{36} + 6\ q^{42} + 21\ q^{46} + 8\ q^{48} + 7\ q^{54} + 7\ q^{56} + 6\ q^{60} + 7\ q^{72} + 6\ q^{78} + q^{84} + 18\ q^{92} + 2\ q^{96} + 7\ q^{108} + q^{120} + 6\ q^{138} + q^{144} + 21\ q^{146} + 6\ q^{152} + 6\ q^{156} + 7\ q^{162} + q^{168} + q^{192} + 6\ q^{198} + q^{216} + 21\ q^{230} + q^{234} + q^{240} + q^{258} + 6\ q^{276} + q^{288} + 2\ q^{312} + 8\ q^{330} + q^{384} + q^{408} + 6\ q^{426} + q^{504}$

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

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).