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Patterns in column strict fillings of rectangular arrays. (English)
PU.M.A., Pure Math. Appl. 22, No. 2, 141-171 (2011).
Summary: In this paper, we study pattern matching in the set $\Cal F_{n,k}$ of fillings of the $k\times n$ rectangle with the integers $1,\dots,kn$ 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 $F \in \Cal F_{n,k}$ 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 $F\in\Cal F_{n,k}$ 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 $\Cal F_{n,k}$ for various classes of standard tableaux of shape $2^k$.
Keywords: pattern; permutation