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Séminaire Lotharingien de Combinatoire, B53h (2006), 22 pp.

# Olga Azenhas and Ricardo Mamede

# Matrix Realizations of Pairs of Young Tableaux, Keys
and Shuffles

**Abstract.**
A key *H* is a semi-standard tableau of partition shape whose
evaluation is a permutation of the shape. We give a necessary and
sufficient condition that the Knuth class of a key equals the set of
shuffles of its columns. In particular, on a three-letters
alphabet the Knuth class of a key equals the set of shuffles of its
columns, and on a four-letters alphabet, the Knuth class of a key is
either the set of shuffles of its columns or the set of shuffles of
its distinct columns with a single word taking appropriate
multiplicities. For some instances of *H*
this result has been already applied to exhibit a matrix
realization, over a local principal ideal domain, of a pair of
tableaux (*T,H*), where *T* is a skew-tableau whose word is
in the Knuth class of *H*. Generalized Lascoux-Sch\"utzenberger
operators, based on nonstandard matching of parentheses, arise
in the matrix realization, over local principal ideal domain, of a pair (*T,H*) on a two-letters alphabet, and they are used
to show that,
over a *t*-letters alphabet, the pair (*T,H*) has a matrix
realization only if the word of *T* is in the Knuth class of
*H*.

Received: December 22, 2004.
Revised: July 24, 2006.
Accepted: August 14, 2006.

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