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Identifier

St000040: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 1

[2,1] => 2

[1,2,3] => 1

[1,3,2] => 2

[2,1,3] => 2

[2,3,1] => 4

[3,1,2] => 4

[3,2,1] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,4,5,1,3] => 28

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,4,1,5,2] => 28

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

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

[3,4,5,1,2] => 46

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

[3,5,1,2,4] => 28

[3,5,1,4,2] => 42

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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 $w$.

References

[1] Hultman, A., Linusson, S., Shareshian, J., Sjöstrand, J. From Bruhat intervals to intersection lattices and a conjecture of Postnikov MathSciNet:2500158
[2] Brewster Lewis, J., Morales, A. H. Combinatorics of diagrams of permutations arXiv:1405.1608
[3] Oh, S., Postnikov, A., Yoo, H. Bruhat order, smooth Schubert varieties, and hyperplane arrangements MathSciNet:2450335
[4] Postnikov, A. Total positivity, Grassmannians, and networks arXiv:math/0609764

Code

def statistic(x):
    # create inversion graph of the permutation
    g = Graph(len(x))
    for inv in x.inversions():
        g.add_edge((inv[0]-1,inv[1]-1))
    # chromatic polynomial of inversion graph
    chrompoly = g.chromatic_polynomial()
    # number of regions of inversion arrangement is the number of
    # acyclic orientations of the inversion graph
    return abs(chrompoly(-1))

Created

Feb 26, 2013 at 06:31 by Alejandro Morales

Updated

Feb 09, 2021 at 14:31 by Martin Rubey