Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 4
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [3,2,1] => [3,1,2] => [3,1,2] => 2
[3,1,2] => [3,2,1] => [3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [3,1,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 6
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 4
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 4
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 8
[2,3,1,4] => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[2,3,4,1] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 6
[2,4,1,3] => [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 10
[2,4,3,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 2
[3,1,2,4] => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 6
[3,2,1,4] => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[3,2,4,1] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 6
[3,4,1,2] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 2
[3,4,2,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 2
[4,1,2,3] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 6
[4,1,3,2] => [4,2,3,1] => [4,3,1,2] => [4,3,1,2] => 6
[4,2,1,3] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 2
[4,2,3,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 2
[4,3,1,2] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 2
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 2
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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
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.