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Identifier

Mp00189: Skew partitions —rotate⟶ Skew partitions

Images

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

[[2],[]] => [[2],[]]

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

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

[[3],[]] => [[3],[]]

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

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

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

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

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

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

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

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

[[4],[]] => [[4],[]]

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

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

[[2,2],[]] => [[2,2],[]]

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

[[4,2],[2]] => [[4,2],[2]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[5],[]] => [[5],[]]

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

[[5,1],[1]] => [[5,4],[4]]

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

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

[[5,2],[2]] => [[5,3],[3]]

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

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

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

[[5,2,1],[2,1]] => [[5,4,3],[4,3]]

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

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

[[5,3],[3]] => [[5,2],[2]]

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

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

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

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

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

[[5,3,1],[3,1]] => [[5,4,2],[4,2]]

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

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

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

[[5,3,2],[3,2]] => [[5,3,2],[3,2]]

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

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

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

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

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

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

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

[[5,3,2,1],[3,2,1]] => [[5,4,3,2],[4,3,2]]

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

[[5,4],[4]] => [[5,1],[1]]

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

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

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

[[5,4,1],[4,1]] => [[5,4,1],[4,1]]

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

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

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

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

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

[[5,4,2],[4,2]] => [[5,3,1],[3,1]]

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

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

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

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

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

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

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

[[5,4,2,1],[4,2,1]] => [[5,4,3,1],[4,3,1]]

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

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

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

[[5,4,3],[4,3]] => [[5,2,1],[2,1]]

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

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

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

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

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

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Description

The rotation of a skew partition.
This is the skew partition obtained by rotating the diagram by 180 degrees. Equivalently, given a skew partition $\lambda/\mu$, its rotation $(\lambda/\mu)^\natural$ is the skew partition with cells $\{(a-i, b-j)| (i, j) \in \lambda/\mu\}$, where $b$ and $a$ are the first part and the number of parts of $\lambda$ respectively.

Properties

graded, involutive

Sage code

def mapping(a):
    if a:
        la = a.outer()
        n = len(la)
        width = la[0]
        nu = [width - p for p in reversed(la)]
        ka = [x + y for x, y in zip(nu, reversed(a.row_lengths()))]
        return SkewPartition([ka, nu])
    return a

Weight

Created

Sep 26, 2020 at 18:34 by Martin Rubey

Updated

Sep 26, 2020 at 18:34 by Martin Rubey