Subsequence Containment by Involutions

Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in Sn which contain a given permutation τ ∈ Sk as a subsequence; this number depends on the patterns of the first j values of τ for 1 ≤ j ≤ k. We then use this to define a partition of Sk, analogous to Wilf-classes in the study of pattern avoidance, and examine properties of this equivalence. In the process, we show that a permutation τ1 . . . τk is layered iff, for 1 ≤ j ≤ k, the pattern of τ1 . . . τj 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.