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Find all semistandard Young tableaux with the same content as a given tableau.

asked 2022-11-11 12:45:11 +0100

lijr07 gravatar image

Let T be a semistandard tableau, for example [[1,3,5],[2,4,9]] (columns). I want to find all semistandard tableaux with the same shape and with the same content as T, which is 1,2,3,4,5,9 (the content can be a multi-set). For example, [[1,2,4],[3,5,9]] (columns) is one of such semistandard tableaux.

I can list all semistandard Young tableaux with maximal number 9:

list(SemistandardTableaux([2,2,2], max_entry=9))

and then select those with the same content as T. But this is slow when T is large. Is there faster way to do this? Thank you very much.

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answered 2022-11-12 03:40:29 +0100

Max Alekseyev gravatar image

updated 2022-11-12 03:42:21 +0100

Here is a quick'n'dirty generator for semistandard tableaux of given shape and with given content:

def gent(shape,content,frontline=None):
    if frontline is None:
        frontline = [0] * len(shape)

    if len(content)==0:
        yield [[0]*s for s in shape]

    e = content[0]
    for row,col in enumerate(frontline):
        if col>=shape[row] or (row>0 and frontline[row-1]<=col):

        frontline[row] += 1

        for t in gent(shape,content[1:],frontline):
            if row+1==len(shape) or col>=shape[row+1] or e<t[row+1][col]:
                t[row][col] = e
                yield t

        frontline[row] -= 1


If given content contains duplicated elements, it may produce the same tableau several times, and so additional filtering may be needed.

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Asked: 2022-11-11 12:45:11 +0100

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Last updated: Nov 12