Subsequence containment by involutions

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Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in Sₙ which contain a given permutation τin Sₖ as a subsequence; this number depends on the patterns of the first j values of τfor 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.