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Séminaire Lotharingien de Combinatoire, B57f (2010), 38 pp.

# Olga Azenhas and Ricardo Mamede

# Matrix Tableau-Pairs with Key and Shuffling Conditions

**Abstract.**
It has been shown that the sequence of Smith invariants defined by
certain sequences of products of matrices, with entries in a local principal
ideal domain, are combinatorially described by tableau-pairs (*T*,*K*)
where *T* is a tableau of skew-shape which rectifies to the
key-tableau *K*. It is a fact that the set of all shuffles of the columns of a key-tableau is a subset of its Knuth class. Here, under the condition that the word of *T* is a
shuffle of the columns of the key-tableau *K*, we show the
converse, that is, every tableau-pair under the aforesaid restrictions has a matrix construction.
In the case of a four-letter alphabet, since we are able to give an explicit description of the Knuth class of a key-tableau as a union of the shuffles of certain subsets of words containing the key-tableau columns, our
construction is general. This may be seen as an indication of a general procedure if a subset of shuffling generators of a generic key-tableau Knuth class is provided. At the moment however, this seems to be a very difficult
problem.

Received: April 23, 2007.
Revised: November 1, 2010.
Accepted: November 12, 2010.

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