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Identifier

St001862: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 0

[1,2] => 0

[1,-2] => 0

[-1,2] => 0

[-1,-2] => 0

[2,1] => 0

[2,-1] => 1

[-2,1] => 0

[-2,-1] => 1

[1,2,3] => 0

[1,2,-3] => 0

[1,-2,3] => 0

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

[-1,2,3] => 0

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

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

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

[1,3,2] => 0

[1,3,-2] => 1

[1,-3,2] => 0

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

[-1,3,2] => 0

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

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

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

[2,1,3] => 0

[2,1,-3] => 0

[2,-1,3] => 1

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

[-2,1,3] => 0

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

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

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

[2,3,1] => 1

[2,3,-1] => 2

[2,-3,1] => 1

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

[-2,3,1] => 1

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

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

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

[3,1,2] => 0

[3,1,-2] => 1

[3,-1,2] => 1

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

[-3,1,2] => 0

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

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

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

[3,2,1] => 0

[3,2,-1] => 1

[3,-2,1] => 2

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

[-3,2,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that

$i < j \leq \pi(i) < \pi(j)$, or
$-i < j \leq -\pi(i) < \pi(j)$, or
$i > j > \pi(i) > \pi(j)$.

References

[1] Corteel, S., Josuat-Vergès, M., Williams, L. K. The Matrix Ansatz, Orthogonal Polynomials, and Permutations arXiv:1005.2696
[2] Corteel, S., Josuat-Vergès, M., Kim, J. S. Crossings of signed permutations and $q$-Eulerian numbers of type $B$ MathSciNet:3096133

Code

def statistic(pi):
    n = len(pi)
    cr1 = sum(1 for j in range(n) for i in range(j) if j + 1 <= pi[i] < pi[j])
    cr2 = sum(1 for j in range(n) for i in range(n) if j + 1 <= -pi[i] < pi[j])
    cr3 = sum(1 for i in range(n) for j in range(i) if j + 1 > pi[i] > pi[j])
    return cr1 + cr2 + cr3

Created

Nov 30, 2022 at 23:44 by Martin Rubey

Updated

Nov 30, 2022 at 23:44 by Martin Rubey