Identifier
Values
[1] => [1,0] => [2,1] => 1
[2] => [1,0,1,0] => [3,1,2] => 2
[1,1] => [1,1,0,0] => [2,3,1] => 2
[3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 4
[1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 3
[4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4
[3,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 5
[2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 6
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 4
[5] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 6
[3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 5
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 7
[2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 4
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 8
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 5
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 6
[3,3] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 5
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 6
[2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 5
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 8
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 7
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 6
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 5
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 5
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 9
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 7
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 5
[] => [] => [1] => 0
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Description
The sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See St000111The sum of the descent tops (or Genocchi descents) of a permutation. and St000154The sum of the descent bottoms of a permutation. for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the drop of a permutation.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.