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] => [1,3,2] => [1,3,2] => [1,3,2] => 4
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 4
[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] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 6
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 4
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 6
[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] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 6
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 6
[2,4,3,1] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 6
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 8
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 4
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 8
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 6
[3,4,2,1] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 6
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 2
[4,1,3,2] => [4,1,3,2] => [3,4,1,2] => [3,4,1,2] => 4
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 4
[4,2,3,1] => [4,1,3,2] => [3,4,1,2] => [3,4,1,2] => 4
[4,3,1,2] => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 8
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 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
Inverse fireworks map
Description
Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
  • $des$ is the number of descents, St000021The number of descents of a permutation.,
  • $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
  • $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
  • $Ebot$ is the sum of the excedance bottoms,
  • $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
  • $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
  • $Res$ is the sum of the (right) embracing numbers,
  • $Ine$ is the sum of the side numbers.