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Identifier

St001907: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 1

[1,2] => 0

[1,-2] => 1

[-1,2] => 1

[-1,-2] => 1

[2,1] => 1

[2,-1] => 2

[-2,1] => 1

[-2,-1] => 1

[1,2,3] => 0

[1,2,-3] => 1

[1,-2,3] => 1

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

[-1,2,3] => 1

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

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

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

[1,3,2] => 1

[1,3,-2] => 2

[1,-3,2] => 1

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

[-1,3,2] => 2

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

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

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

[2,1,3] => 1

[2,1,-3] => 2

[2,-1,3] => 2

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

[-2,1,3] => 1

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

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

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

[2,3,1] => 2

[2,3,-1] => 3

[2,-3,1] => 2

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

[-2,3,1] => 2

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

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

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

[3,1,2] => 1

[3,1,-2] => 2

[3,-1,2] => 2

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

[-3,1,2] => 1

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

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

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

[3,2,1] => 1

[3,2,-1] => 2

[3,-2,1] => 2

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

[-3,2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1 + q$

$F_{2} = 1 + 6\ q + q^{2}$

$F_{3} = 1 + 23\ q + 23\ q^{2} + q^{3}$

$F_{4} = 1 + 76\ q + 230\ q^{2} + 76\ q^{3} + q^{4}$

Description

The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation.
For a signed permutation

$\sigma$ , this equals

$\left\lfloor \dfrac{fexc(\sigma)+1}{2} \right\rfloor = exc(\sigma) + \left\lfloor \dfrac{neg(\sigma)+1}{2} \right\rfloor,$
where

$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$

$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|,$

$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|.$
This statistic has the same distribution as the descent statistic St001427The number of descents of a signed permutation..

References

[1] Bastidas, J. The polytope algebra of generalized permutahedra MathSciNet:4339357
[2] Bastidas, J., Hohlweg, C., Saliola, F. The Primitive Eulerian polynomial arXiv:2306.15556

Code

def statistic(w):
    exc = sum(1 for i in range(len(w)) if w[i] > i+1)
    neg = sum(1 for i in range(len(w)) if w[i] < 0)
    return (2*exc + neg + 1) // 2

Created

Jul 05, 2023 at 17:47 by Jose Bastidas

Updated

Jul 06, 2023 at 15:10 by Martin Rubey