Identifier
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [2,3,1] => [3,2,1] => [3,2,1] => 2
[1,0,1,1,0,0] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0] => [3,1,2] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 2
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 2
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [4,3,1,2] => [4,3,1,2] => 2
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 2
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 2
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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.