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]).
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
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