Partial matchings and pattern avoidance

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


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 vn(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 vn(τ) is always rational for a certain infinite family of patterns τ. We also provide some general explicit formulas for vn(τ) in terms of vn(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.

Original languageEnglish
Pages (from-to)25-50
Number of pages26
JournalApplicable Analysis and Discrete Mathematics
Issue number1
StatePublished - Apr 2013


  • Involution
  • Kernel method
  • Pattern avoidance
  • Set partition

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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