Subsequence Containment by Involutions

Abstract

Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in ${\cal S}_{n}$ which contain a given permutation $\tau\in{\cal S}_{k}$ as a subsequence; this number depends on the patterns of the first $j$ values of $\tau$ for $1\leq j\leq k$. We then use this to define a partition of ${\cal S}_{k}$, analogous to Wilf-classes in the study of pattern avoidance, and examine properties of this equivalence. In the process, we show that a permutation $\tau_1\ldots\tau_k$ is layered iff, for $1\leq j\leq k$, the pattern of $\tau_1\ldots\tau_j$ is an involution. We also obtain a result of Sagan and Stanley counting the standard Young tableaux of size $n$ which contain a fixed tableau of size $k$ as a subtableau.