Partial matchings and pattern avoidance

Abstract

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 characterized by the avoidance of a single pattern, specializing a natural notion of partition containment that has been introduced by Sagan. Let vn(?) denote the number of partial matchings of size n which avoid the pattern ?. Among our results, we show that the generating function for the numbers vn(?) is always rational for a certain infinite family of patterns ?. We also provide some general explicit formulas for vn(?) in terms of vn(p), where p is a pattern contained in ?. 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.