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Identifier

St001433: Signed permutations ⟶ ℤ

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

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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