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Identifier
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 [-1,3,-2,-4]=>7 [-1,-3,2,4]=>4 [-1,-3,2,-4]=>8 [-1,-3,-2,4]=>5 [-1,-3,-2,-4]=>9 [1,3,4,2]=>1 [1,3,4,-2]=>3 [1,3,-4,2]=>4 [1,3,-4,-2]=>6 [1,-3,4,2]=>3 [1,-3,4,-2]=>5 [1,-3,-4,2]=>6 [1,-3,-4,-2]=>8 [-1,3,4,2]=>2 [-1,3,4,-2]=>4 [-1,3,-4,2]=>5 [-1,3,-4,-2]=>7 [-1,-3,4,2]=>4 [-1,-3,4,-2]=>6 [-1,-3,-4,2]=>7 [-1,-3,-4,-2]=>9 [1,4,2,3]=>1 [1,4,2,-3]=>5 [1,4,-2,3]=>2 [1,4,-2,-3]=>6 [1,-4,2,3]=>3 [1,-4,2,-3]=>7 [1,-4,-2,3]=>4 [1,-4,-2,-3]=>8 [-1,4,2,3]=>2 [-1,4,2,-3]=>6 [-1,4,-2,3]=>3 [-1,4,-2,-3]=>7 [-1,-4,2,3]=>4 [-1,-4,2,-3]=>8 [-1,-4,-2,3]=>5 [-1,-4,-2,-3]=>9 [1,4,3,2]=>2 [1,4,3,-2]=>4 [1,4,-3,2]=>3 [1,4,-3,-2]=>5 [1,-4,3,2]=>4 [1,-4,3,-2]=>6 [1,-4,-3,2]=>5 [1,-4,-3,-2]=>7 [-1,4,3,2]=>3 [-1,4,3,-2]=>5 [-1,4,-3,2]=>4 [-1,4,-3,-2]=>6 [-1,-4,3,2]=>5 [-1,-4,3,-2]=>7 [-1,-4,-3,2]=>6 [-1,-4,-3,-2]=>8 [2,1,3,4]=>1 [2,1,3,-4]=>5 [2,1,-3,4]=>4 [2,1,-3,-4]=>8 [2,-1,3,4]=>1 [2,-1,3,-4]=>5 [2,-1,-3,4]=>4 [2,-1,-3,-4]=>8 [-2,1,3,4]=>2 [-2,1,3,-4]=>6 [-2,1,-3,4]=>5 [-2,1,-3,-4]=>9 [-2,-1,3,4]=>2 [-2,-1,3,-4]=>6 [-2,-1,-3,4]=>5 [-2,-1,-3,-4]=>9 [2,1,4,3]=>2 [2,1,4,-3]=>4 [2,1,-4,3]=>5 [2,1,-4,-3]=>7 [2,-1,4,3]=>2 [2,-1,4,-3]=>4 [2,-1,-4,3]=>5 [2,-1,-4,-3]=>7 [-2,1,4,3]=>3 [-2,1,4,-3]=>5 [-2,1,-4,3]=>6 [-2,1,-4,-3]=>8 [-2,-1,4,3]=>3 [-2,-1,4,-3]=>5 [-2,-1,-4,3]=>6 [-2,-1,-4,-3]=>8 [2,3,1,4]=>1 [2,3,1,-4]=>5 [2,3,-1,4]=>2 [2,3,-1,-4]=>6 [2,-3,1,4]=>3 [2,-3,1,-4]=>7 [2,-3,-1,4]=>4 [2,-3,-1,-4]=>8 [-2,3,1,4]=>2 [-2,3,1,-4]=>6 [-2,3,-1,4]=>3 [-2,3,-1,-4]=>7 [-2,-3,1,4]=>4 [-2,-3,1,-4]=>8 [-2,-3,-1,4]=>5 [-2,-3,-1,-4]=>9 [2,3,4,1]=>2 [2,3,4,-1]=>2 [2,3,-4,1]=>5 [2,3,-4,-1]=>5 [2,-3,4,1]=>4 [2,-3,4,-1]=>4 [2,-3,-4,1]=>7 [2,-3,-4,-1]=>7 [-2,3,4,1]=>3 [-2,3,4,-1]=>3 [-2,3,-4,1]=>6 [-2,3,-4,-1]=>6 [-2,-3,4,1]=>5 [-2,-3,4,-1]=>5 [-2,-3,-4,1]=>8 [-2,-3,-4,-1]=>8 [2,4,1,3]=>1 [2,4,1,-3]=>5 [2,4,-1,3]=>2 [2,4,-1,-3]=>6 [2,-4,1,3]=>3 [2,-4,1,-3]=>7 [2,-4,-1,3]=>4 [2,-4,-1,-3]=>8 [-2,4,1,3]=>2 [-2,4,1,-3]=>6 [-2,4,-1,3]=>3 [-2,4,-1,-3]=>7 [-2,-4,1,3]=>4 [-2,-4,1,-3]=>8 [-2,-4,-1,3]=>5 [-2,-4,-1,-3]=>9 [2,4,3,1]=>3 [2,4,3,-1]=>3 [2,4,-3,1]=>4 [2,4,-3,-1]=>4 [2,-4,3,1]=>5 [2,-4,3,-1]=>5 [2,-4,-3,1]=>6 [2,-4,-3,-1]=>6 [-2,4,3,1]=>4 [-2,4,3,-1]=>4 [-2,4,-3,1]=>5 [-2,4,-3,-1]=>5 [-2,-4,3,1]=>6 [-2,-4,3,-1]=>6 [-2,-4,-3,1]=>7 [-2,-4,-3,-1]=>7 [3,1,2,4]=>1 [3,1,2,-4]=>5 [3,1,-2,4]=>4 [3,1,-2,-4]=>8 [3,-1,2,4]=>1 [3,-1,2,-4]=>5 [3,-1,-2,4]=>4 [3,-1,-2,-4]=>8 [-3,1,2,4]=>2 [-3,1,2,-4]=>6 [-3,1,-2,4]=>5 [-3,1,-2,-4]=>9 [-3,-1,2,4]=>2 [-3,-1,2,-4]=>6 [-3,-1,-2,4]=>5 [-3,-1,-2,-4]=>9 [3,1,4,2]=>3 [3,1,4,-2]=>3 [3,1,-4,2]=>6 [3,1,-4,-2]=>6 [3,-1,4,2]=>3 [3,-1,4,-2]=>3 [3,-1,-4,2]=>6 [3,-1,-4,-2]=>6 [-3,1,4,2]=>4 [-3,1,4,-2]=>4 [-3,1,-4,2]=>7 [-3,1,-4,-2]=>7 [-3,-1,4,2]=>4 [-3,-1,4,-2]=>4 [-3,-1,-4,2]=>7 [-3,-1,-4,-2]=>7 [3,2,1,4]=>2 [3,2,1,-4]=>6 [3,2,-1,4]=>3 [3,2,-1,-4]=>7 [3,-2,1,4]=>2 [3,-2,1,-4]=>6 [3,-2,-1,4]=>3 [3,-2,-1,-4]=>7 [-3,2,1,4]=>3 [-3,2,1,-4]=>7 [-3,2,-1,4]=>4 [-3,2,-1,-4]=>8 [-3,-2,1,4]=>3 [-3,-2,1,-4]=>7 [-3,-2,-1,4]=>4 [-3,-2,-1,-4]=>8 [3,2,4,1]=>3 [3,2,4,-1]=>3 [3,2,-4,1]=>6 [3,2,-4,-1]=>6 [3,-2,4,1]=>3 [3,-2,4,-1]=>3 [3,-2,-4,1]=>6 [3,-2,-4,-1]=>6 [-3,2,4,1]=>4 [-3,2,4,-1]=>4 [-3,2,-4,1]=>7 [-3,2,-4,-1]=>7 [-3,-2,4,1]=>4 [-3,-2,4,-1]=>4 [-3,-2,-4,1]=>7 [-3,-2,-4,-1]=>7 [3,4,1,2]=>2 [3,4,1,-2]=>4 [3,4,-1,2]=>3 [3,4,-1,-2]=>5 [3,-4,1,2]=>4 [3,-4,1,-2]=>6 [3,-4,-1,2]=>5 [3,-4,-1,-2]=>7 [-3,4,1,2]=>3 [-3,4,1,-2]=>5 [-3,4,-1,2]=>4 [-3,4,-1,-2]=>6 [-3,-4,1,2]=>5 [-3,-4,1,-2]=>7 [-3,-4,-1,2]=>6 [-3,-4,-1,-2]=>8 [3,4,2,1]=>3 [3,4,2,-1]=>3 [3,4,-2,1]=>4 [3,4,-2,-1]=>4 [3,-4,2,1]=>5 [3,-4,2,-1]=>5 [3,-4,-2,1]=>6 [3,-4,-2,-1]=>6 [-3,4,2,1]=>4 [-3,4,2,-1]=>4 [-3,4,-2,1]=>5 [-3,4,-2,-1]=>5 [-3,-4,2,1]=>6 [-3,-4,2,-1]=>6 [-3,-4,-2,1]=>7 [-3,-4,-2,-1]=>7 [4,1,2,3]=>2 [4,1,2,-3]=>4 [4,1,-2,3]=>5 [4,1,-2,-3]=>7 [4,-1,2,3]=>2 [4,-1,2,-3]=>4 [4,-1,-2,3]=>5 [4,-1,-2,-3]=>7 [-4,1,2,3]=>3 [-4,1,2,-3]=>5 [-4,1,-2,3]=>6 [-4,1,-2,-3]=>8 [-4,-1,2,3]=>3 [-4,-1,2,-3]=>5 [-4,-1,-2,3]=>6 [-4,-1,-2,-3]=>8 [4,1,3,2]=>3 [4,1,3,-2]=>3 [4,1,-3,2]=>6 [4,1,-3,-2]=>6 [4,-1,3,2]=>3 [4,-1,3,-2]=>3 [4,-1,-3,2]=>6 [4,-1,-3,-2]=>6 [-4,1,3,2]=>4 [-4,1,3,-2]=>4 [-4,1,-3,2]=>7 [-4,1,-3,-2]=>7 [-4,-1,3,2]=>4 [-4,-1,3,-2]=>4 [-4,-1,-3,2]=>7 [-4,-1,-3,-2]=>7 [4,2,1,3]=>3 [4,2,1,-3]=>5 [4,2,-1,3]=>4 [4,2,-1,-3]=>6 [4,-2,1,3]=>3 [4,-2,1,-3]=>5 [4,-2,-1,3]=>4 [4,-2,-1,-3]=>6 [-4,2,1,3]=>4 [-4,2,1,-3]=>6 [-4,2,-1,3]=>5 [-4,2,-1,-3]=>7 [-4,-2,1,3]=>4 [-4,-2,1,-3]=>6 [-4,-2,-1,3]=>5 [-4,-2,-1,-3]=>7 [4,2,3,1]=>3 [4,2,3,-1]=>3 [4,2,-3,1]=>6 [4,2,-3,-1]=>6 [4,-2,3,1]=>3 [4,-2,3,-1]=>3 [4,-2,-3,1]=>6 [4,-2,-3,-1]=>6 [-4,2,3,1]=>4 [-4,2,3,-1]=>4 [-4,2,-3,1]=>7 [-4,2,-3,-1]=>7 [-4,-2,3,1]=>4 [-4,-2,3,-1]=>4 [-4,-2,-3,1]=>7 [-4,-2,-3,-1]=>7 [4,3,1,2]=>3 [4,3,1,-2]=>5 [4,3,-1,2]=>4 [4,3,-1,-2]=>6 [4,-3,1,2]=>3 [4,-3,1,-2]=>5 [4,-3,-1,2]=>4 [4,-3,-1,-2]=>6 [-4,3,1,2]=>4 [-4,3,1,-2]=>6 [-4,3,-1,2]=>5 [-4,3,-1,-2]=>7 [-4,-3,1,2]=>4 [-4,-3,1,-2]=>6 [-4,-3,-1,2]=>5 [-4,-3,-1,-2]=>7 [4,3,2,1]=>4 [4,3,2,-1]=>4 [4,3,-2,1]=>5 [4,3,-2,-1]=>5 [4,-3,2,1]=>4 [4,-3,2,-1]=>4 [4,-3,-2,1]=>5 [4,-3,-2,-1]=>5 [-4,3,2,1]=>5 [-4,3,2,-1]=>5 [-4,3,-2,1]=>6 [-4,3,-2,-1]=>6 [-4,-3,2,1]=>5 [-4,-3,2,-1]=>5 [-4,-3,-2,1]=>6 [-4,-3,-2,-1]=>6
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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