Identifier
Values
[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [1,2] => 0
[1,1] => [[1],[2]] => [2,1] => [2,1] => 1
[3] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[2,1] => [[1,3],[2]] => [2,1,3] => [2,1,3] => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [2,3,1] => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => 1
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [4,1,3,2] => 2
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [2,3,1,4] => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [2,3,4,1] => 1
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [4,1,3,2,5] => 2
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [5,4,2,1,3] => 3
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [2,3,4,1,5] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [2,3,4,5,1] => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [4,1,3,2,5,6] => 2
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [2,3,1,4,5,6] => 1
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [6,1,2,4,5,3] => 2
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [5,4,2,1,3,6] => 3
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [2,3,4,1,5,6] => 1
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [4,1,6,3,5,2] => 3
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [6,3,5,2,1,4] => 4
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [2,3,4,5,1,6] => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [2,3,4,5,6,1] => 1
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
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Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of St000021The number of descents of a permutation. and St000354The number of recoils of a permutation..
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
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.