- FindStat

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Identifier

Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux

Images

{{1}} => [[1]]

{{1,2}} => [[1,2]]

{{1},{2}} => [[1],[2]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5,6}} => [[1,2,3,4,5,6]]

{{1,2,3,4,5},{6}} => [[1,2,3,4,5],[6]]

{{1,2,3,4,6},{5}} => [[1,2,3,4,6],[5]]

{{1,2,3,4},{5,6}} => [[1,2,3,4],[5,6]]

{{1,2,3,4},{5},{6}} => [[1,2,3,4],[5],[6]]

{{1,2,3,5,6},{4}} => [[1,2,3,5,6],[4]]

{{1,2,3,5},{4,6}} => [[1,2,3,5],[4,6]]

{{1,2,3,5},{4},{6}} => [[1,2,3,5],[4],[6]]

{{1,2,3,6},{4,5}} => [[1,2,3,6],[4,5]]

{{1,2,3},{4,5,6}} => [[1,2,3],[4,5,6]]

{{1,2,3},{4,5},{6}} => [[1,2,3],[4,5],[6]]

{{1,2,3,6},{4},{5}} => [[1,2,3,6],[4],[5]]

{{1,2,3},{4,6},{5}} => [[1,2,3],[4,6],[5]]

{{1,2,3},{4},{5,6}} => [[1,2,3],[4,6],[5]]

{{1,2,3},{4},{5},{6}} => [[1,2,3],[4],[5],[6]]

{{1,2,4,5,6},{3}} => [[1,2,4,5,6],[3]]

{{1,2,4,5},{3,6}} => [[1,2,4,5],[3,6]]

{{1,2,4,5},{3},{6}} => [[1,2,4,5],[3],[6]]

{{1,2,4,6},{3,5}} => [[1,2,4,6],[3,5]]

{{1,2,4},{3,5,6}} => [[1,2,4],[3,5,6]]

{{1,2,4},{3,5},{6}} => [[1,2,4],[3,5],[6]]

{{1,2,4,6},{3},{5}} => [[1,2,4,6],[3],[5]]

{{1,2,4},{3,6},{5}} => [[1,2,4],[3,6],[5]]

{{1,2,4},{3},{5,6}} => [[1,2,4],[3,6],[5]]

{{1,2,4},{3},{5},{6}} => [[1,2,4],[3],[5],[6]]

{{1,2,5,6},{3,4}} => [[1,2,5,6],[3,4]]

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Description

Sends a set partition to the associated standard tableau.
The $j$th column of the standard tableau associated to a set partition is the set of $j$th smallest elements of its blocks arranged in increassing order.

References

[1] Prasad, A., Ram, S. Set partitions, tableaux, and subspace profiles under regular split semisimple matrices arXiv:2112.00479

Properties

surjective

Sage code

def mapping(x):
    """
    Return the standard tableau associated to a set partition.
    
    EXAMPLE::

    sage: mapping([[1, 2], [3, 6, 7], [4, 5]])
    [[1, 2, 7], [3, 5], [4, 6]]
    """
    A = [sorted(a) for a in x]
    m = max(len(a) for a in A)
    if m==1:
        return Tableau(sorted(a for a in A))
    else:
        last_row = sorted(a[-1] for a in A if len(a)==m)
        l = len(last_row)
        AA = [a[:m-1] for a in A]
        TT = mapping(AA)
        T = []
        for i,r in enumerate(TT):
            if i

Weight

Created

Jul 20, 2022 at 16:52 by Amritanshu Prasad

Updated

Jul 20, 2022 at 16:52 by Amritanshu Prasad