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Identifier
Values
=>
[1]=>0 [-1]=>1 [1,2]=>0 [1,-2]=>3 [-1,2]=>1 [-1,-2]=>4 [2,1]=>2 [2,-1]=>3 [-2,1]=>1 [-2,-1]=>2 [1,2,3]=>0 [1,2,-3]=>5 [1,-2,3]=>3 [1,-2,-3]=>8 [-1,2,3]=>1 [-1,2,-3]=>6 [-1,-2,3]=>4 [-1,-2,-3]=>9 [1,3,2]=>4 [1,3,-2]=>5 [1,-3,2]=>3 [1,-3,-2]=>4 [-1,3,2]=>5 [-1,3,-2]=>6 [-1,-3,2]=>4 [-1,-3,-2]=>5 [2,1,3]=>2 [2,1,-3]=>7 [2,-1,3]=>3 [2,-1,-3]=>8 [-2,1,3]=>1 [-2,1,-3]=>6 [-2,-1,3]=>2 [-2,-1,-3]=>7 [2,3,1]=>4 [2,3,-1]=>5 [2,-3,1]=>3 [2,-3,-1]=>4 [-2,3,1]=>5 [-2,3,-1]=>6 [-2,-3,1]=>4 [-2,-3,-1]=>5 [3,1,2]=>2 [3,1,-2]=>7 [3,-1,2]=>3 [3,-1,-2]=>8 [-3,1,2]=>1 [-3,1,-2]=>6 [-3,-1,2]=>2 [-3,-1,-2]=>7 [3,2,1]=>6 [3,2,-1]=>7 [3,-2,1]=>3 [3,-2,-1]=>4 [-3,2,1]=>5 [-3,2,-1]=>6 [-3,-2,1]=>2 [-3,-2,-1]=>3 [1,2,3,4]=>0 [1,2,3,-4]=>7 [1,2,-3,4]=>5 [1,2,-3,-4]=>12 [1,-2,3,4]=>3 [1,-2,3,-4]=>10 [1,-2,-3,4]=>8 [1,-2,-3,-4]=>15 [-1,2,3,4]=>1 [-1,2,3,-4]=>8 [-1,2,-3,4]=>6 [-1,2,-3,-4]=>13 [-1,-2,3,4]=>4 [-1,-2,3,-4]=>11 [-1,-2,-3,4]=>9 [-1,-2,-3,-4]=>16 [1,2,4,3]=>6 [1,2,4,-3]=>7 [1,2,-4,3]=>5 [1,2,-4,-3]=>6 [1,-2,4,3]=>9 [1,-2,4,-3]=>10 [1,-2,-4,3]=>8 [1,-2,-4,-3]=>9 [-1,2,4,3]=>7 [-1,2,4,-3]=>8 [-1,2,-4,3]=>6 [-1,2,-4,-3]=>7 [-1,-2,4,3]=>10 [-1,-2,4,-3]=>11 [-1,-2,-4,3]=>9 [-1,-2,-4,-3]=>10 [1,3,2,4]=>4 [1,3,2,-4]=>11 [1,3,-2,4]=>5 [1,3,-2,-4]=>12 [1,-3,2,4]=>3 [1,-3,2,-4]=>10 [1,-3,-2,4]=>4 [1,-3,-2,-4]=>11 [-1,3,2,4]=>5 [-1,3,2,-4]=>12 [-1,3,-2,4]=>6 [-1,3,-2,-4]=>13 [-1,-3,2,4]=>4 [-1,-3,2,-4]=>11 [-1,-3,-2,4]=>5 [-1,-3,-2,-4]=>12 [1,3,4,2]=>6 [1,3,4,-2]=>7 [1,3,-4,2]=>5 [1,3,-4,-2]=>6 [1,-3,4,2]=>9 [1,-3,4,-2]=>10 [1,-3,-4,2]=>8 [1,-3,-4,-2]=>9 [-1,3,4,2]=>7 [-1,3,4,-2]=>8 [-1,3,-4,2]=>6 [-1,3,-4,-2]=>7 [-1,-3,4,2]=>10 [-1,-3,4,-2]=>11 [-1,-3,-4,2]=>9 [-1,-3,-4,-2]=>10 [1,4,2,3]=>4 [1,4,2,-3]=>11 [1,4,-2,3]=>5 [1,4,-2,-3]=>12 [1,-4,2,3]=>3 [1,-4,2,-3]=>10 [1,-4,-2,3]=>4 [1,-4,-2,-3]=>11 [-1,4,2,3]=>5 [-1,4,2,-3]=>12 [-1,4,-2,3]=>6 [-1,4,-2,-3]=>13 [-1,-4,2,3]=>4 [-1,-4,2,-3]=>11 [-1,-4,-2,3]=>5 [-1,-4,-2,-3]=>12 [1,4,3,2]=>10 [1,4,3,-2]=>11 [1,4,-3,2]=>5 [1,4,-3,-2]=>6 [1,-4,3,2]=>9 [1,-4,3,-2]=>10 [1,-4,-3,2]=>4 [1,-4,-3,-2]=>5 [-1,4,3,2]=>11 [-1,4,3,-2]=>12 [-1,4,-3,2]=>6 [-1,4,-3,-2]=>7 [-1,-4,3,2]=>10 [-1,-4,3,-2]=>11 [-1,-4,-3,2]=>5 [-1,-4,-3,-2]=>6 [2,1,3,4]=>2 [2,1,3,-4]=>9 [2,1,-3,4]=>7 [2,1,-3,-4]=>14 [2,-1,3,4]=>3 [2,-1,3,-4]=>10 [2,-1,-3,4]=>8 [2,-1,-3,-4]=>15 [-2,1,3,4]=>1 [-2,1,3,-4]=>8 [-2,1,-3,4]=>6 [-2,1,-3,-4]=>13 [-2,-1,3,4]=>2 [-2,-1,3,-4]=>9 [-2,-1,-3,4]=>7 [-2,-1,-3,-4]=>14 [2,1,4,3]=>8 [2,1,4,-3]=>9 [2,1,-4,3]=>7 [2,1,-4,-3]=>8 [2,-1,4,3]=>9 [2,-1,4,-3]=>10 [2,-1,-4,3]=>8 [2,-1,-4,-3]=>9 [-2,1,4,3]=>7 [-2,1,4,-3]=>8 [-2,1,-4,3]=>6 [-2,1,-4,-3]=>7 [-2,-1,4,3]=>8 [-2,-1,4,-3]=>9 [-2,-1,-4,3]=>7 [-2,-1,-4,-3]=>8 [2,3,1,4]=>4 [2,3,1,-4]=>11 [2,3,-1,4]=>5 [2,3,-1,-4]=>12 [2,-3,1,4]=>3 [2,-3,1,-4]=>10 [2,-3,-1,4]=>4 [2,-3,-1,-4]=>11 [-2,3,1,4]=>5 [-2,3,1,-4]=>12 [-2,3,-1,4]=>6 [-2,3,-1,-4]=>13 [-2,-3,1,4]=>4 [-2,-3,1,-4]=>11 [-2,-3,-1,4]=>5 [-2,-3,-1,-4]=>12 [2,3,4,1]=>6 [2,3,4,-1]=>7 [2,3,-4,1]=>5 [2,3,-4,-1]=>6 [2,-3,4,1]=>9 [2,-3,4,-1]=>10 [2,-3,-4,1]=>8 [2,-3,-4,-1]=>9 [-2,3,4,1]=>7 [-2,3,4,-1]=>8 [-2,3,-4,1]=>6 [-2,3,-4,-1]=>7 [-2,-3,4,1]=>10 [-2,-3,4,-1]=>11 [-2,-3,-4,1]=>9 [-2,-3,-4,-1]=>10 [2,4,1,3]=>4 [2,4,1,-3]=>11 [2,4,-1,3]=>5 [2,4,-1,-3]=>12 [2,-4,1,3]=>3 [2,-4,1,-3]=>10 [2,-4,-1,3]=>4 [2,-4,-1,-3]=>11 [-2,4,1,3]=>5 [-2,4,1,-3]=>12 [-2,4,-1,3]=>6 [-2,4,-1,-3]=>13 [-2,-4,1,3]=>4 [-2,-4,1,-3]=>11 [-2,-4,-1,3]=>5 [-2,-4,-1,-3]=>12 [2,4,3,1]=>10 [2,4,3,-1]=>11 [2,4,-3,1]=>5 [2,4,-3,-1]=>6 [2,-4,3,1]=>9 [2,-4,3,-1]=>10 [2,-4,-3,1]=>4 [2,-4,-3,-1]=>5 [-2,4,3,1]=>11 [-2,4,3,-1]=>12 [-2,4,-3,1]=>6 [-2,4,-3,-1]=>7 [-2,-4,3,1]=>10 [-2,-4,3,-1]=>11 [-2,-4,-3,1]=>5 [-2,-4,-3,-1]=>6 [3,1,2,4]=>2 [3,1,2,-4]=>9 [3,1,-2,4]=>7 [3,1,-2,-4]=>14 [3,-1,2,4]=>3 [3,-1,2,-4]=>10 [3,-1,-2,4]=>8 [3,-1,-2,-4]=>15 [-3,1,2,4]=>1 [-3,1,2,-4]=>8 [-3,1,-2,4]=>6 [-3,1,-2,-4]=>13 [-3,-1,2,4]=>2 [-3,-1,2,-4]=>9 [-3,-1,-2,4]=>7 [-3,-1,-2,-4]=>14 [3,1,4,2]=>8 [3,1,4,-2]=>9 [3,1,-4,2]=>7 [3,1,-4,-2]=>8 [3,-1,4,2]=>9 [3,-1,4,-2]=>10 [3,-1,-4,2]=>8 [3,-1,-4,-2]=>9 [-3,1,4,2]=>7 [-3,1,4,-2]=>8 [-3,1,-4,2]=>6 [-3,1,-4,-2]=>7 [-3,-1,4,2]=>8 [-3,-1,4,-2]=>9 [-3,-1,-4,2]=>7 [-3,-1,-4,-2]=>8 [3,2,1,4]=>6 [3,2,1,-4]=>13 [3,2,-1,4]=>7 [3,2,-1,-4]=>14 [3,-2,1,4]=>3 [3,-2,1,-4]=>10 [3,-2,-1,4]=>4 [3,-2,-1,-4]=>11 [-3,2,1,4]=>5 [-3,2,1,-4]=>12 [-3,2,-1,4]=>6 [-3,2,-1,-4]=>13 [-3,-2,1,4]=>2 [-3,-2,1,-4]=>9 [-3,-2,-1,4]=>3 [-3,-2,-1,-4]=>10 [3,2,4,1]=>8 [3,2,4,-1]=>9 [3,2,-4,1]=>7 [3,2,-4,-1]=>8 [3,-2,4,1]=>9 [3,-2,4,-1]=>10 [3,-2,-4,1]=>8 [3,-2,-4,-1]=>9 [-3,2,4,1]=>7 [-3,2,4,-1]=>8 [-3,2,-4,1]=>6 [-3,2,-4,-1]=>7 [-3,-2,4,1]=>8 [-3,-2,4,-1]=>9 [-3,-2,-4,1]=>7 [-3,-2,-4,-1]=>8 [3,4,1,2]=>4 [3,4,1,-2]=>11 [3,4,-1,2]=>5 [3,4,-1,-2]=>12 [3,-4,1,2]=>3 [3,-4,1,-2]=>10 [3,-4,-1,2]=>4 [3,-4,-1,-2]=>11 [-3,4,1,2]=>5 [-3,4,1,-2]=>12 [-3,4,-1,2]=>6 [-3,4,-1,-2]=>13 [-3,-4,1,2]=>4 [-3,-4,1,-2]=>11 [-3,-4,-1,2]=>5 [-3,-4,-1,-2]=>12 [3,4,2,1]=>10 [3,4,2,-1]=>11 [3,4,-2,1]=>5 [3,4,-2,-1]=>6 [3,-4,2,1]=>9 [3,-4,2,-1]=>10 [3,-4,-2,1]=>4 [3,-4,-2,-1]=>5 [-3,4,2,1]=>11 [-3,4,2,-1]=>12 [-3,4,-2,1]=>6 [-3,4,-2,-1]=>7 [-3,-4,2,1]=>10 [-3,-4,2,-1]=>11 [-3,-4,-2,1]=>5 [-3,-4,-2,-1]=>6 [4,1,2,3]=>2 [4,1,2,-3]=>9 [4,1,-2,3]=>7 [4,1,-2,-3]=>14 [4,-1,2,3]=>3 [4,-1,2,-3]=>10 [4,-1,-2,3]=>8 [4,-1,-2,-3]=>15 [-4,1,2,3]=>1 [-4,1,2,-3]=>8 [-4,1,-2,3]=>6 [-4,1,-2,-3]=>13 [-4,-1,2,3]=>2 [-4,-1,2,-3]=>9 [-4,-1,-2,3]=>7 [-4,-1,-2,-3]=>14 [4,1,3,2]=>8 [4,1,3,-2]=>9 [4,1,-3,2]=>7 [4,1,-3,-2]=>8 [4,-1,3,2]=>9 [4,-1,3,-2]=>10 [4,-1,-3,2]=>8 [4,-1,-3,-2]=>9 [-4,1,3,2]=>7 [-4,1,3,-2]=>8 [-4,1,-3,2]=>6 [-4,1,-3,-2]=>7 [-4,-1,3,2]=>8 [-4,-1,3,-2]=>9 [-4,-1,-3,2]=>7 [-4,-1,-3,-2]=>8 [4,2,1,3]=>6 [4,2,1,-3]=>13 [4,2,-1,3]=>7 [4,2,-1,-3]=>14 [4,-2,1,3]=>3 [4,-2,1,-3]=>10 [4,-2,-1,3]=>4 [4,-2,-1,-3]=>11 [-4,2,1,3]=>5 [-4,2,1,-3]=>12 [-4,2,-1,3]=>6 [-4,2,-1,-3]=>13 [-4,-2,1,3]=>2 [-4,-2,1,-3]=>9 [-4,-2,-1,3]=>3 [-4,-2,-1,-3]=>10 [4,2,3,1]=>8 [4,2,3,-1]=>9 [4,2,-3,1]=>7 [4,2,-3,-1]=>8 [4,-2,3,1]=>9 [4,-2,3,-1]=>10 [4,-2,-3,1]=>8 [4,-2,-3,-1]=>9 [-4,2,3,1]=>7 [-4,2,3,-1]=>8 [-4,2,-3,1]=>6 [-4,2,-3,-1]=>7 [-4,-2,3,1]=>8 [-4,-2,3,-1]=>9 [-4,-2,-3,1]=>7 [-4,-2,-3,-1]=>8 [4,3,1,2]=>6 [4,3,1,-2]=>13 [4,3,-1,2]=>7 [4,3,-1,-2]=>14 [4,-3,1,2]=>3 [4,-3,1,-2]=>10 [4,-3,-1,2]=>4 [4,-3,-1,-2]=>11 [-4,3,1,2]=>5 [-4,3,1,-2]=>12 [-4,3,-1,2]=>6 [-4,3,-1,-2]=>13 [-4,-3,1,2]=>2 [-4,-3,1,-2]=>9 [-4,-3,-1,2]=>3 [-4,-3,-1,-2]=>10 [4,3,2,1]=>12 [4,3,2,-1]=>13 [4,3,-2,1]=>7 [4,3,-2,-1]=>8 [4,-3,2,1]=>9 [4,-3,2,-1]=>10 [4,-3,-2,1]=>4 [4,-3,-2,-1]=>5 [-4,3,2,1]=>11 [-4,3,2,-1]=>12 [-4,3,-2,1]=>6 [-4,3,-2,-1]=>7 [-4,-3,2,1]=>8 [-4,-3,2,-1]=>9 [-4,-3,-2,1]=>3 [-4,-3,-2,-1]=>4
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Description
The flag major index of a signed permutation.
The flag major index of a signed permutation $\sigma$ is:
$$\operatorname{fmaj}(\sigma)=\operatorname{neg}(\sigma)+2\cdot \sum_{i\in \operatorname{Des}_B(\sigma)}{i} ,$$
where $\operatorname{Des}_B(\sigma)$ is the $B$-descent set of $\sigma$; see [1, Eq.(10)].
This statistic is equidistributed with the $B$-inversions (St001428The number of B-inversions of a signed permutation.) and with the negative major index on the groups of signed permutations (see [1, Corollary 4.6]).
References
[1] Adinh, R. M., Brenti, F., Roichman, Y. Descent numbers and major indices for the hyperoctahedral group MathSciNet:2003m:05211 arXiv:math/0012111
Code
def statistic(pi):
    pi = [0] + list(pi)
    return  (sum(1 for i in range(len(pi)) if pi[i] < 0)
            + sum(2*i for i in range(len(pi)-1) if pi[i] > pi[i+1]))

Created
Jun 24, 2019 at 15:21 by Angela Carnevale
Updated
Jan 08, 2023 at 15:10 by Martin Rubey