Identifier
Values
[[1,2]] => [1,2] => [1,2] => [1,2] => 1
[[1],[2]] => [2,1] => [2,1] => [2,1] => 0
[[1,2,3]] => [1,2,3] => [1,2,3] => [1,2,3] => 2
[[1,3],[2]] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[[1,2],[3]] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[[1],[2],[3]] => [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3
[[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[1,2,4],[3]] => [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[[1,2,3],[4]] => [4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 2
[[1,3],[2,4]] => [2,4,1,3] => [4,2,1,3] => [2,1,4,3] => 0
[[1,2],[3,4]] => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 0
[[1,2],[3],[4]] => [4,3,1,2] => [3,1,4,2] => [1,3,4,2] => 2
[[1],[2],[3],[4]] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 2
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2
[[1,2,4,5],[3]] => [3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 2
[[1,2,3,5],[4]] => [4,1,2,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => 2
[[1,2,3,4],[5]] => [5,1,2,3,4] => [5,1,2,3,4] => [1,2,3,5,4] => 3
[[1,3,5],[2,4]] => [2,4,1,3,5] => [4,2,1,3,5] => [2,1,4,3,5] => 0
[[1,2,5],[3,4]] => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => 0
[[1,3,4],[2,5]] => [2,5,1,3,4] => [5,2,1,3,4] => [2,1,3,5,4] => 1
[[1,2,4],[3,5]] => [3,5,1,2,4] => [5,1,3,2,4] => [3,1,2,5,4] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => 1
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => 2
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [3,1,4,2,5] => [1,3,4,2,5] => 2
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [2,5,1,3,4] => [2,1,5,3,4] => 0
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => 0
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [4,1,2,5,3] => [1,2,4,5,3] => 3
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [5,3,2,1,4] => [3,2,1,5,4] => 0
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [5,4,2,1,3] => [2,1,5,4,3] => 0
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [5,1,4,3,2] => [5,4,1,3,2] => 0
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [4,5,2,1,3] => [4,2,1,5,3] => 0
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [4,1,5,3,2] => [4,5,1,3,2] => 1
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => 2
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [2,3,5,1,4] => [5,2,3,1,4] => 0
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [2,4,1,5,3] => [4,2,5,1,3] => 0
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [3,1,4,5,2] => [1,3,4,5,2] => 3
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => 3
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Description
The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$.
A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
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.