Partial matchings and pattern avoidance

A partition of a finite set all of whose blocks have size one or two is called a partial matching. Here, we enumerate classes of partial matchings charac-terized by the avoidance of a single pattern, specializing a natural notion of partition containment that has been introduced by Sagan. Let v_n(τ) denote the number of partial matchings of size n which avoid the pattern τ. Among our results, we show that the generating function for the numbers v_n(τ) is always rational for a certain infinite family of patterns τ. We also provide some general explicit formulas for v_n(τ) in terms of v_n(ρ), where ρ is a pat-tern contained in τ. Finally, we find, with two exceptions, explicit formulas and/or generating functions for the number of partial matchings avoiding any pattern of length at most five.