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Identifier

St001870: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 1

[1,2] => 0

[1,-2] => 2

[-1,2] => 1

[-1,-2] => 1

[2,1] => 0

[2,-1] => 2

[-2,1] => 1

[-2,-1] => 1

[1,2,3] => 0

[1,2,-3] => 2

[1,-2,3] => 2

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

[-1,2,3] => 1

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

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

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

[1,3,2] => 0

[1,3,-2] => 2

[1,-3,2] => 2

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

[-1,3,2] => 1

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

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

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

[2,1,3] => 0

[2,1,-3] => 2

[2,-1,3] => 2

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

[-2,1,3] => 1

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

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

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

[2,3,1] => 0

[2,3,-1] => 2

[2,-3,1] => 2

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

[-2,3,1] => 1

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

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

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

[3,1,2] => 0

[3,1,-2] => 2

[3,-1,2] => 2

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

[-3,1,2] => 1

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

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

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

[3,2,1] => 0

[3,2,-1] => 2

[3,-2,1] => 2

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

[-3,2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.

References

[1] 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):
    return sum(1 for i in range(-len(pi), len(pi)) if (i != -1 and i != 0 and pi(i) > 0 and pi(i+1) < 0) or (i == -1 and pi(-1) > 0 and pi(1) < 0))

Created

Dec 01, 2022 at 16:31 by Martin Rubey

Updated

Dec 01, 2022 at 16:31 by Martin Rubey