Enumeration of Matchings: Problems and Progress

TL;DR

This paper reviews the history, progress, and open problems in the enumeration of matchings, highlighting advances made since 1996 and presenting new challenges for future research.

Contribution

It provides a comprehensive overview of 32 problems in matching enumeration, including solutions, progress, and new open problems since 1996.

Findings

Half of the original problems were solved by 1999

Progress was made through collaborative efforts and literature review

The paper introduces a dozen new open problems in the field

Abstract

This document is built around a list of thirty-two problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and on-line literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems. (Note: This article supersedes math.CO/9801060 and math.CO/9801061.)