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Identifier

St001823: Signed permutations ⟶ ℤ

Values

[1] => 0

[-1] => 1

[1,2] => 0

[1,-2] => 2

[-1,2] => 1

[-1,-2] => 3

[2,1] => 1

[2,-1] => 1

[-2,1] => 2

[-2,-1] => 2

[1,2,3] => 0

[1,2,-3] => 3

[1,-2,3] => 2

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

[-1,2,3] => 1

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

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

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

[1,3,2] => 1

[1,3,-2] => 2

[1,-3,2] => 3

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

[-1,3,2] => 2

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

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

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

[2,1,3] => 1

[2,1,-3] => 4

[2,-1,3] => 1

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

[-2,1,3] => 2

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

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

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

[2,3,1] => 1

[2,3,-1] => 2

[2,-3,1] => 3

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

[-2,3,1] => 2

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

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

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

[3,1,2] => 1

[3,1,-2] => 4

[3,-1,2] => 1

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

[-3,1,2] => 2

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

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

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

[3,2,1] => 2

[3,2,-1] => 3

[3,-2,1] => 2

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

[-3,2,1] => 3

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

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

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

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

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

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

[1,2,-3,-4] => 7

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

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

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

[1,-2,-3,-4] => 9

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

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

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

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

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

[-1,-2,3,-4] => 7

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

[-1,-2,-3,-4] => 10

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

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

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

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

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

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

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

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

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

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

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

[-1,2,-4,-3] => 7

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

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

[-1,-2,-4,3] => 7

[-1,-2,-4,-3] => 9

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

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

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

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

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

[1,-3,2,-4] => 7

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

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

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

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

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

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Description

The Stasinski-Voll length of a signed permutation.
The Stasinski-Voll length of a signed permutation $\sigma$ is
$$ L(\sigma) = \frac{1}{2} \#\{(i,j) ~\mid -n \leq i < j \leq n,~ i \not\equiv j \operatorname{mod} 2,~ \sigma(i) > \sigma(j)\}, $$
where $n$ is the size of $\sigma$.

References

[1] Stasinski, A., Voll, C. A new statistic on the hyperoctahedral groups arXiv:1303.0990

Code

def statistic(pi):
    n = pi.parent().rank()
    x = [-i for i in list(pi)]
    x.reverse()
    w = x + [0] + list(pi)
    
    l = sum(1 for i in range(-n,n+1) for j in range(i+1,n+1) if w[i+n] > w[j+n] and i%2 != j%2)
    
    return l/2

Created

Jul 22, 2022 at 11:45 by Dennis Jahn

Updated

Jul 22, 2022 at 11:45 by Dennis Jahn