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Identifier
Values
[1] => 0
[-1] => 1
[1,2] => 0
[1,-2] => 1
[-1,2] => 1
[-1,-2] => 2
[2,1] => 2
[2,-1] => 3
[-2,1] => 3
[-2,-1] => 4
[1,2,3] => 0
[1,2,-3] => 1
[1,-2,3] => 1
[1,-2,-3] => 2
[-1,2,3] => 1
[-1,2,-3] => 2
[-1,-2,3] => 2
[-1,-2,-3] => 3
[1,3,2] => 4
[1,3,-2] => 5
[1,-3,2] => 5
[1,-3,-2] => 6
[-1,3,2] => 5
[-1,3,-2] => 6
[-1,-3,2] => 6
[-1,-3,-2] => 7
[2,1,3] => 2
[2,1,-3] => 3
[2,-1,3] => 3
[2,-1,-3] => 4
[-2,1,3] => 3
[-2,1,-3] => 4
[-2,-1,3] => 4
[-2,-1,-3] => 5
[2,3,1] => 6
[2,3,-1] => 7
[2,-3,1] => 7
[2,-3,-1] => 8
[-2,3,1] => 7
[-2,3,-1] => 8
[-2,-3,1] => 8
[-2,-3,-1] => 9
[3,1,2] => 2
[3,1,-2] => 3
[3,-1,2] => 3
[3,-1,-2] => 4
[-3,1,2] => 3
[-3,1,-2] => 4
[-3,-1,2] => 4
[-3,-1,-2] => 5
[3,2,1] => 4
[3,2,-1] => 5
[3,-2,1] => 5
[3,-2,-1] => 6
[-3,2,1] => 5
[-3,2,-1] => 6
[-3,-2,1] => 6
[-3,-2,-1] => 7
[1,2,3,4] => 0
[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] => 3
[-1,2,3,4] => 1
[-1,2,3,-4] => 2
[-1,2,-3,4] => 2
[-1,2,-3,-4] => 3
[-1,-2,3,4] => 2
[-1,-2,3,-4] => 3
[-1,-2,-3,4] => 3
[-1,-2,-3,-4] => 4
[1,2,4,3] => 6
[1,2,4,-3] => 7
[1,2,-4,3] => 7
[1,2,-4,-3] => 8
[1,-2,4,3] => 7
[1,-2,4,-3] => 8
[1,-2,-4,3] => 8
[1,-2,-4,-3] => 9
[-1,2,4,3] => 7
[-1,2,4,-3] => 8
[-1,2,-4,3] => 8
[-1,2,-4,-3] => 9
[-1,-2,4,3] => 8
[-1,-2,4,-3] => 9
[-1,-2,-4,3] => 9
[-1,-2,-4,-3] => 10
[1,3,2,4] => 4
[1,3,2,-4] => 5
[1,3,-2,4] => 5
[1,3,-2,-4] => 6
[1,-3,2,4] => 5
[1,-3,2,-4] => 6
[1,-3,-2,4] => 6
[1,-3,-2,-4] => 7
[-1,3,2,4] => 5
[-1,3,2,-4] => 6
[-1,3,-2,4] => 6
>>> Load all 442 entries. <<<
[-1,3,-2,-4] => 7
[-1,-3,2,4] => 6
[-1,-3,2,-4] => 7
[-1,-3,-2,4] => 7
[-1,-3,-2,-4] => 8
[1,3,4,2] => 10
[1,3,4,-2] => 11
[1,3,-4,2] => 11
[1,3,-4,-2] => 12
[1,-3,4,2] => 11
[1,-3,4,-2] => 12
[1,-3,-4,2] => 12
[1,-3,-4,-2] => 13
[-1,3,4,2] => 11
[-1,3,4,-2] => 12
[-1,3,-4,2] => 12
[-1,3,-4,-2] => 13
[-1,-3,4,2] => 12
[-1,-3,4,-2] => 13
[-1,-3,-4,2] => 13
[-1,-3,-4,-2] => 14
[1,4,2,3] => 4
[1,4,2,-3] => 5
[1,4,-2,3] => 5
[1,4,-2,-3] => 6
[1,-4,2,3] => 5
[1,-4,2,-3] => 6
[1,-4,-2,3] => 6
[1,-4,-2,-3] => 7
[-1,4,2,3] => 5
[-1,4,2,-3] => 6
[-1,4,-2,3] => 6
[-1,4,-2,-3] => 7
[-1,-4,2,3] => 6
[-1,-4,2,-3] => 7
[-1,-4,-2,3] => 7
[-1,-4,-2,-3] => 8
[1,4,3,2] => 6
[1,4,3,-2] => 7
[1,4,-3,2] => 7
[1,4,-3,-2] => 8
[1,-4,3,2] => 7
[1,-4,3,-2] => 8
[1,-4,-3,2] => 8
[1,-4,-3,-2] => 9
[-1,4,3,2] => 7
[-1,4,3,-2] => 8
[-1,4,-3,2] => 8
[-1,4,-3,-2] => 9
[-1,-4,3,2] => 8
[-1,-4,3,-2] => 9
[-1,-4,-3,2] => 9
[-1,-4,-3,-2] => 10
[2,1,3,4] => 2
[2,1,3,-4] => 3
[2,1,-3,4] => 3
[2,1,-3,-4] => 4
[2,-1,3,4] => 3
[2,-1,3,-4] => 4
[2,-1,-3,4] => 4
[2,-1,-3,-4] => 5
[-2,1,3,4] => 3
[-2,1,3,-4] => 4
[-2,1,-3,4] => 4
[-2,1,-3,-4] => 5
[-2,-1,3,4] => 4
[-2,-1,3,-4] => 5
[-2,-1,-3,4] => 5
[-2,-1,-3,-4] => 6
[2,1,4,3] => 8
[2,1,4,-3] => 9
[2,1,-4,3] => 9
[2,1,-4,-3] => 10
[2,-1,4,3] => 9
[2,-1,4,-3] => 10
[2,-1,-4,3] => 10
[2,-1,-4,-3] => 11
[-2,1,4,3] => 9
[-2,1,4,-3] => 10
[-2,1,-4,3] => 10
[-2,1,-4,-3] => 11
[-2,-1,4,3] => 10
[-2,-1,4,-3] => 11
[-2,-1,-4,3] => 11
[-2,-1,-4,-3] => 12
[2,3,1,4] => 6
[2,3,1,-4] => 7
[2,3,-1,4] => 7
[2,3,-1,-4] => 8
[2,-3,1,4] => 7
[2,-3,1,-4] => 8
[2,-3,-1,4] => 8
[2,-3,-1,-4] => 9
[-2,3,1,4] => 7
[-2,3,1,-4] => 8
[-2,3,-1,4] => 8
[-2,3,-1,-4] => 9
[-2,-3,1,4] => 8
[-2,-3,1,-4] => 9
[-2,-3,-1,4] => 9
[-2,-3,-1,-4] => 10
[2,3,4,1] => 12
[2,3,4,-1] => 13
[2,3,-4,1] => 13
[2,3,-4,-1] => 14
[2,-3,4,1] => 13
[2,-3,4,-1] => 14
[2,-3,-4,1] => 14
[2,-3,-4,-1] => 15
[-2,3,4,1] => 13
[-2,3,4,-1] => 14
[-2,3,-4,1] => 14
[-2,3,-4,-1] => 15
[-2,-3,4,1] => 14
[-2,-3,4,-1] => 15
[-2,-3,-4,1] => 15
[-2,-3,-4,-1] => 16
[2,4,1,3] => 6
[2,4,1,-3] => 7
[2,4,-1,3] => 7
[2,4,-1,-3] => 8
[2,-4,1,3] => 7
[2,-4,1,-3] => 8
[2,-4,-1,3] => 8
[2,-4,-1,-3] => 9
[-2,4,1,3] => 7
[-2,4,1,-3] => 8
[-2,4,-1,3] => 8
[-2,4,-1,-3] => 9
[-2,-4,1,3] => 8
[-2,-4,1,-3] => 9
[-2,-4,-1,3] => 9
[-2,-4,-1,-3] => 10
[2,4,3,1] => 8
[2,4,3,-1] => 9
[2,4,-3,1] => 9
[2,4,-3,-1] => 10
[2,-4,3,1] => 9
[2,-4,3,-1] => 10
[2,-4,-3,1] => 10
[2,-4,-3,-1] => 11
[-2,4,3,1] => 9
[-2,4,3,-1] => 10
[-2,4,-3,1] => 10
[-2,4,-3,-1] => 11
[-2,-4,3,1] => 10
[-2,-4,3,-1] => 11
[-2,-4,-3,1] => 11
[-2,-4,-3,-1] => 12
[3,1,2,4] => 2
[3,1,2,-4] => 3
[3,1,-2,4] => 3
[3,1,-2,-4] => 4
[3,-1,2,4] => 3
[3,-1,2,-4] => 4
[3,-1,-2,4] => 4
[3,-1,-2,-4] => 5
[-3,1,2,4] => 3
[-3,1,2,-4] => 4
[-3,1,-2,4] => 4
[-3,1,-2,-4] => 5
[-3,-1,2,4] => 4
[-3,-1,2,-4] => 5
[-3,-1,-2,4] => 5
[-3,-1,-2,-4] => 6
[3,1,4,2] => 8
[3,1,4,-2] => 9
[3,1,-4,2] => 9
[3,1,-4,-2] => 10
[3,-1,4,2] => 9
[3,-1,4,-2] => 10
[3,-1,-4,2] => 10
[3,-1,-4,-2] => 11
[-3,1,4,2] => 9
[-3,1,4,-2] => 10
[-3,1,-4,2] => 10
[-3,1,-4,-2] => 11
[-3,-1,4,2] => 10
[-3,-1,4,-2] => 11
[-3,-1,-4,2] => 11
[-3,-1,-4,-2] => 12
[3,2,1,4] => 4
[3,2,1,-4] => 5
[3,2,-1,4] => 5
[3,2,-1,-4] => 6
[3,-2,1,4] => 5
[3,-2,1,-4] => 6
[3,-2,-1,4] => 6
[3,-2,-1,-4] => 7
[-3,2,1,4] => 5
[-3,2,1,-4] => 6
[-3,2,-1,4] => 6
[-3,2,-1,-4] => 7
[-3,-2,1,4] => 6
[-3,-2,1,-4] => 7
[-3,-2,-1,4] => 7
[-3,-2,-1,-4] => 8
[3,2,4,1] => 10
[3,2,4,-1] => 11
[3,2,-4,1] => 11
[3,2,-4,-1] => 12
[3,-2,4,1] => 11
[3,-2,4,-1] => 12
[3,-2,-4,1] => 12
[3,-2,-4,-1] => 13
[-3,2,4,1] => 11
[-3,2,4,-1] => 12
[-3,2,-4,1] => 12
[-3,2,-4,-1] => 13
[-3,-2,4,1] => 12
[-3,-2,4,-1] => 13
[-3,-2,-4,1] => 13
[-3,-2,-4,-1] => 14
[3,4,1,2] => 6
[3,4,1,-2] => 7
[3,4,-1,2] => 7
[3,4,-1,-2] => 8
[3,-4,1,2] => 7
[3,-4,1,-2] => 8
[3,-4,-1,2] => 8
[3,-4,-1,-2] => 9
[-3,4,1,2] => 7
[-3,4,1,-2] => 8
[-3,4,-1,2] => 8
[-3,4,-1,-2] => 9
[-3,-4,1,2] => 8
[-3,-4,1,-2] => 9
[-3,-4,-1,2] => 9
[-3,-4,-1,-2] => 10
[3,4,2,1] => 8
[3,4,2,-1] => 9
[3,4,-2,1] => 9
[3,4,-2,-1] => 10
[3,-4,2,1] => 9
[3,-4,2,-1] => 10
[3,-4,-2,1] => 10
[3,-4,-2,-1] => 11
[-3,4,2,1] => 9
[-3,4,2,-1] => 10
[-3,4,-2,1] => 10
[-3,4,-2,-1] => 11
[-3,-4,2,1] => 10
[-3,-4,2,-1] => 11
[-3,-4,-2,1] => 11
[-3,-4,-2,-1] => 12
[4,1,2,3] => 2
[4,1,2,-3] => 3
[4,1,-2,3] => 3
[4,1,-2,-3] => 4
[4,-1,2,3] => 3
[4,-1,2,-3] => 4
[4,-1,-2,3] => 4
[4,-1,-2,-3] => 5
[-4,1,2,3] => 3
[-4,1,2,-3] => 4
[-4,1,-2,3] => 4
[-4,1,-2,-3] => 5
[-4,-1,2,3] => 4
[-4,-1,2,-3] => 5
[-4,-1,-2,3] => 5
[-4,-1,-2,-3] => 6
[4,1,3,2] => 4
[4,1,3,-2] => 5
[4,1,-3,2] => 5
[4,1,-3,-2] => 6
[4,-1,3,2] => 5
[4,-1,3,-2] => 6
[4,-1,-3,2] => 6
[4,-1,-3,-2] => 7
[-4,1,3,2] => 5
[-4,1,3,-2] => 6
[-4,1,-3,2] => 6
[-4,1,-3,-2] => 7
[-4,-1,3,2] => 6
[-4,-1,3,-2] => 7
[-4,-1,-3,2] => 7
[-4,-1,-3,-2] => 8
[4,2,1,3] => 4
[4,2,1,-3] => 5
[4,2,-1,3] => 5
[4,2,-1,-3] => 6
[4,-2,1,3] => 5
[4,-2,1,-3] => 6
[4,-2,-1,3] => 6
[4,-2,-1,-3] => 7
[-4,2,1,3] => 5
[-4,2,1,-3] => 6
[-4,2,-1,3] => 6
[-4,2,-1,-3] => 7
[-4,-2,1,3] => 6
[-4,-2,1,-3] => 7
[-4,-2,-1,3] => 7
[-4,-2,-1,-3] => 8
[4,2,3,1] => 6
[4,2,3,-1] => 7
[4,2,-3,1] => 7
[4,2,-3,-1] => 8
[4,-2,3,1] => 7
[4,-2,3,-1] => 8
[4,-2,-3,1] => 8
[4,-2,-3,-1] => 9
[-4,2,3,1] => 7
[-4,2,3,-1] => 8
[-4,2,-3,1] => 8
[-4,2,-3,-1] => 9
[-4,-2,3,1] => 8
[-4,-2,3,-1] => 9
[-4,-2,-3,1] => 9
[-4,-2,-3,-1] => 10
[4,3,1,2] => 8
[4,3,1,-2] => 9
[4,3,-1,2] => 9
[4,3,-1,-2] => 10
[4,-3,1,2] => 9
[4,-3,1,-2] => 10
[4,-3,-1,2] => 10
[4,-3,-1,-2] => 11
[-4,3,1,2] => 9
[-4,3,1,-2] => 10
[-4,3,-1,2] => 10
[-4,3,-1,-2] => 11
[-4,-3,1,2] => 10
[-4,-3,1,-2] => 11
[-4,-3,-1,2] => 11
[-4,-3,-1,-2] => 12
[4,3,2,1] => 10
[4,3,2,-1] => 11
[4,3,-2,1] => 11
[4,3,-2,-1] => 12
[4,-3,2,1] => 11
[4,-3,2,-1] => 12
[4,-3,-2,1] => 12
[4,-3,-2,-1] => 13
[-4,3,2,1] => 11
[-4,3,2,-1] => 12
[-4,3,-2,1] => 12
[-4,3,-2,-1] => 13
[-4,-3,2,1] => 12
[-4,-3,2,-1] => 13
[-4,-3,-2,1] => 13
[-4,-3,-2,-1] => 14
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Description
The flag Denert index of a signed permutation.
The flag Denert index of a signed permutation $\sigma$ is:
$$fden(\sigma) = \operatorname{neg}(\sigma) + 2 \cdot den_B(\sigma),$$
where $den_B(\sigma) = den(perm(\sigma))$ is the Denert index of the associated permutation to $\sigma$.
References
[1] Fire, M. Statistics on Wreath Products arXiv:math/0409421
Code
def statistic(pi):
    n = pi.parent().rank()
    
    A = sum(1 for l,k in Subsets(range(n),2) if abs(pi[k]) < abs(pi[l]) <= k+1 )
    B = sum(1 for l,k in Subsets(range(n),2) if abs(pi[l]) <= k+1 < abs(pi[k]) )
    C = sum(1 for l,k in Subsets(range(n),2) if k+1 < abs(pi[k]) < abs(pi[l])  )
    den_B = A+B+C
    
    neg = len([i for i in pi if i < 0])
    
    return neg + 2 * den_B
Created
Jul 20, 2022 at 08:55 by Dennis Jahn
Updated
Jul 20, 2022 at 08:55 by Dennis Jahn