Identifier
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
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
Jun 24, 2019 at 15:21 by Angela Carnevale