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Identifier

Mp00187: Skew partitions —conjugate⟶ Skew partitions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The conjugate of the skew partition.
The conjugate of a skew partition $\lambda$ is the skew partition $\lambda^*$ whose Ferrers diagram is obtained from the Ferrers diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.

Properties

bijective, involutive

Sage code

def mapping(la):
    return la.conjugate()

Weight

Created

Sep 24, 2020 at 20:09 by Martin Rubey

Updated

Sep 24, 2020 at 20:09 by Martin Rubey