**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]).

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/0012111Code

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**searching the database

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