Patterns in column strict fillings of rectangular arrays

Abstract

In this paper, we study pattern matching in the set F_{n,k} of fillings of the k x n rectangle with the integers 1,...,kn such that the elements in any column increase from bottom to top. Let P be a column strict tableau of shape 2^k. We say that a filling F in F_{n,k} has P-match starting at i if the elements of F in columns i and i+1 have the same relative order as the elements of P. We compute the generating functions for the distribution of P-matches and nonoverlapping P-matches for various classes of standard tableaux of shape 2^k. We say that a filling F in F_{n,k} is P-alternating if there are P-matches of F starting at all odd positions but there are no P-matches of F starting at even positions. We also compute the generating functions for P-alternating elements of F_{n,k} for various classes of standard tableaux of shape 2^k.