Matrix Tableau-pairs with Key and Shuffling Conditions

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.