Identifier
Values
[1] => [1,0] => [[1],[2]] => [2,1] => 1
[2] => [1,0,1,0] => [[1,3],[2,4]] => [2,4,1,3] => 4
[1,1] => [1,1,0,0] => [[1,2],[3,4]] => [3,4,1,2] => 5
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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
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
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.