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] => 8
[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] => 6
[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] => 11
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 13
[3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 6
[2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 12
[2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
[] => [] => [1] => 0
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Description
The reduced word complexity of a permutation.
For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$.
For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$.
This statistic appears in [1, Question 6.1].
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.