Propp's benzels and Lai's nearly symmetric hexagons with holes

TL;DR

This paper introduces a new factorization theorem for symmetric graphs, solving open problems in tiling enumeration and providing simplified formulas for perfect matchings in complex regions.

Contribution

It presents a novel factorization theorem for symmetric graph matchings and applies it to solve open problems in tiling enumeration and perfect matchings.

Findings

Solved four open problems of Propp on trimer tilings.

Derived exact enumeration formulas for hexagonal regions with holes.

Provided new, simpler proofs for known perfect matching formulas.

Abstract

In this paper we present a new version of the second author's factorization theorem for perfect matchings of symmetric graphs. We then use our result to solve four open problems of Propp on the enumeration of trimer tilings on the hexagonal lattice. As another application, we obtain a semi-factorization result for the number of lozenge tilings of a large class of hexagonal regions with holes (obtained by starting with an arbitrary symmetric hexagon with holes, and translating all the holes one unit lattice segment in the same direction). This in turn leads to the solution of two open problems posed by Lai and to an extension of a result due to Fulmek and Krattenthaler, which results in exact enumeration formulas for some new families of hexagonal regions with holes. Our result also allows us to find new, simpler proofs (and in one case, a new, simpler form) of some formulas due to Krattenthaler for the number of perfect matchings of Aztec rectangles with unit holes along a lattice diagonal.