Perfect matchings and perfect powers

TL;DR

This paper introduces a unified approach to prove that certain graph families have the number of perfect matchings as perfect powers or twice perfect powers, confirming several conjectures and providing new enumeration results.

Contribution

It presents a unified method for proving perfect power matchings in graphs, confirming conjectures, and deriving new multivariate enumeration formulas.

Findings

Proved that Aztec dungeon regions have tilings as powers or twice powers of 13.

Confirmed that certain subgraphs have perfect matchings as powers or twice powers of 3.

Developed multivariate enumeration formulas for Aztec diamonds and fortress graphs.

Abstract

In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers. In this paper we present an approach that allows proving them in a unified way. We use this approach to prove a conjecture of James Propp stating that the number of tilings of the so-called Aztec dungeon regions is a power (or twice a power) of 13. We also prove a conjecture of Matt Blum stating that the number of perfect matchings of a certain family of subgraphs of the square lattice is a power of 3 or twice a power of 3. In addition we obtain multi-parameter generalizations of previously known results, and new multi-parameter exact enumeration results. We obtain in particular a simple combinatorial proof of Bo-Yin Yang's multivariate generalization of fortresses, a result whose previously known proof was quite complicated, amounting to evaluation of the Kasteleyn matrix by explicit row reduction. We also include a new multivariate exact enumeration of Aztec diamonds, in the spirit of Stanley's multivariate version.