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Identifier

Mp00233: Dyck paths —skew partition⟶ Skew partitions

Images

[1,0] => [[1],[]]

[1,0,1,0] => [[1,1],[]]

[1,1,0,0] => [[2],[]]

[1,0,1,0,1,0] => [[1,1,1],[]]

[1,0,1,1,0,0] => [[2,1],[]]

[1,1,0,0,1,0] => [[2,2],[1]]

[1,1,0,1,0,0] => [[3],[]]

[1,1,1,0,0,0] => [[2,2],[]]

[1,0,1,0,1,0,1,0] => [[1,1,1,1],[]]

[1,0,1,0,1,1,0,0] => [[2,1,1],[]]

[1,0,1,1,0,0,1,0] => [[2,2,1],[1]]

[1,0,1,1,0,1,0,0] => [[3,1],[]]

[1,0,1,1,1,0,0,0] => [[2,2,1],[]]

[1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]]

[1,1,0,0,1,1,0,0] => [[3,2],[1]]

[1,1,0,1,0,0,1,0] => [[3,3],[2]]

[1,1,0,1,0,1,0,0] => [[4],[]]

[1,1,0,1,1,0,0,0] => [[3,3],[1]]

[1,1,1,0,0,0,1,0] => [[2,2,2],[1]]

[1,1,1,0,0,1,0,0] => [[3,2],[]]

[1,1,1,0,1,0,0,0] => [[2,2,2],[]]

[1,1,1,1,0,0,0,0] => [[3,3],[]]

[1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]]

[1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]]

[1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]]

[1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]]

[1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1],[]]

[1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]]

[1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]]

[1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]]

[1,0,1,1,0,1,0,1,0,0] => [[4,1],[]]

[1,0,1,1,0,1,1,0,0,0] => [[3,3,1],[1]]

[1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1],[1]]

[1,0,1,1,1,0,0,1,0,0] => [[3,2,1],[]]

[1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1],[]]

[1,0,1,1,1,1,0,0,0,0] => [[3,3,1],[]]

[1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]]

[1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]]

[1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]]

[1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]]

[1,1,0,0,1,1,1,0,0,0] => [[3,3,2],[1,1]]

[1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]]

[1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]]

[1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]]

[1,1,0,1,0,1,0,1,0,0] => [[5],[]]

[1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]]

[1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]]

[1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]]

[1,1,0,1,1,0,1,0,0,0] => [[3,3,3],[1,1]]

[1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]]

[1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2],[1,1]]

[1,1,1,0,0,0,1,1,0,0] => [[3,2,2],[1]]

[1,1,1,0,0,1,0,0,1,0] => [[3,3,2],[2]]

[1,1,1,0,0,1,0,1,0,0] => [[4,2],[]]

[1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]]

[1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2],[1]]

[1,1,1,0,1,0,0,1,0,0] => [[3,2,2],[]]

[1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2],[]]

[1,1,1,0,1,1,0,0,0,0] => [[3,3,2],[]]

[1,1,1,1,0,0,0,0,1,0] => [[3,3,3],[2]]

[1,1,1,1,0,0,0,1,0,0] => [[4,3],[]]

[1,1,1,1,0,0,1,0,0,0] => [[3,3,3],[1]]

[1,1,1,1,0,1,0,0,0,0] => [[4,4],[]]

[1,1,1,1,1,0,0,0,0,0] => [[3,3,3],[]]

[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1],[]]

[1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1],[]]

[1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1],[1]]

[1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1],[]]

[1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1],[]]

[1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1],[1,1]]

[1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1],[1]]

[1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1],[2]]

[1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1],[]]

[1,0,1,0,1,1,0,1,1,0,0,0] => [[3,3,1,1],[1]]

[1,0,1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1,1],[1]]

[1,0,1,0,1,1,1,0,0,1,0,0] => [[3,2,1,1],[]]

[1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1],[]]

[1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1],[]]

[1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1],[1,1,1]]

[1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1],[1,1]]

[1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1],[2,1]]

[1,0,1,1,0,0,1,1,0,1,0,0] => [[4,2,1],[1]]

[1,0,1,1,0,0,1,1,1,0,0,0] => [[3,3,2,1],[1,1]]

[1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1],[2,2]]

[1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1],[2]]

[1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1],[3]]

[1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1],[]]

[1,0,1,1,0,1,0,1,1,0,0,0] => [[4,4,1],[2]]

[1,0,1,1,0,1,1,0,0,0,1,0] => [[3,3,3,1],[2,1]]

[1,0,1,1,0,1,1,0,0,1,0,0] => [[4,3,1],[1]]

[1,0,1,1,0,1,1,0,1,0,0,0] => [[3,3,3,1],[1,1]]

[1,0,1,1,0,1,1,1,0,0,0,0] => [[4,4,1],[1]]

[1,0,1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2,1],[1,1]]

[1,0,1,1,1,0,0,0,1,1,0,0] => [[3,2,2,1],[1]]

[1,0,1,1,1,0,0,1,0,0,1,0] => [[3,3,2,1],[2]]

[1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1],[]]

[1,0,1,1,1,0,0,1,1,0,0,0] => [[3,3,2,1],[1]]

[1,0,1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2,1],[1]]

[1,0,1,1,1,0,1,0,0,1,0,0] => [[3,2,2,1],[]]

[1,0,1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2,1],[]]

[1,0,1,1,1,0,1,1,0,0,0,0] => [[3,3,2,1],[]]

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Description

The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.

Properties

injective

Sage code

def mapping(D):
    p = ParallelogramPolyomino.from_dyck_word(D)
    return SkewPartitions(p.area()).from_row_and_column_length(p.widths()[::-1], p.heights())

Weight

Created

May 13, 2021 at 15:12 by Martin Rubey

Updated

May 13, 2021 at 15:12 by Martin Rubey