## Enumeration of Matchings: Problems and Progress (1999)

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Venue: | in New Perspectives in Algebraic Combinatorics |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Propp99enumerationof,

author = {James Propp},

title = {Enumeration of Matchings: Problems and Progress},

booktitle = {in New Perspectives in Algebraic Combinatorics},

year = {1999},

pages = {255--291},

publisher = {University Press}

}

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### 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. 1. Introduction How many perfect matchings does a given graph G have? That is, in how many ways can one choose a subset of the edges of G so that each vertex of G belongs to one and only one chosen edge? (See Figure 1(a) for an example of a perfect matching of a graph.) For general graph...