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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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* This information is distributed in the hope that it will be useful, *
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-----------------------------------------------------------------------------
Statistic identifier: St001433
-----------------------------------------------------------------------------
Collection: Signed permutations
-----------------------------------------------------------------------------
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 ([[St001428]]) 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]))
-----------------------------------------------------------------------------
Statistic 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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Created: Jun 24, 2019 at 15:21 by Angela Carnevale
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Last Updated: Jan 08, 2023 at 15:10 by Martin Rubey