Identifier
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [-2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [-3,1,2] => 1
[2,1,3] => [2,1,3] => [-2,1,3] => 1
[2,3,1] => [2,3,1] => [-3,-2,1] => 2
[3,1,2] => [3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [2,-3,1] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [-4,-3,1,2] => 2
[1,4,2,3] => [1,4,2,3] => [4,1,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => 3
[2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => 2
[2,3,1,4] => [2,3,1,4] => [-3,-2,1,4] => 2
[2,3,4,1] => [2,3,4,1] => [-4,-3,-2,1] => 3
[2,4,1,3] => [2,4,1,3] => [2,-4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => [3,-4,-2,1] => 4
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [3,1,4,2] => [4,-3,1,2] => 3
[3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => 3
[3,2,4,1] => [3,2,4,1] => [2,-4,-3,1] => 4
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 4
[3,4,2,1] => [3,4,2,1] => [2,4,-3,1] => 5
[4,1,2,3] => [4,1,2,3] => [1,-4,2,3] => 3
[4,1,3,2] => [4,1,3,2] => [-3,-4,1,2] => 4
[4,2,1,3] => [4,2,1,3] => [-2,-4,1,3] => 4
[4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => 5
[4,3,1,2] => [4,3,1,2] => [-4,3,1,2] => 5
[4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => 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]).
Map
Foata-Han inverse
Description
The inverse of the Foata-Han bijection for signed permutations.
This map sends the the flag inversion number St001428The number of B-inversions of a signed permutation. to the flag major index St001433The flag major index of a signed permutation.
Map
to signed permutation
Description
The signed permutation with all signs positive.