Periodic lozenge tilings of the plane

TL;DR

This paper studies the enumeration of doubly periodic lozenge tilings of the plane using Pfaffian methods, introduces an alternative computation approach, and proposes new classes of tilings as open problems.

Contribution

It presents a new method for counting doubly periodic lozenge tilings and explores additional tiling classes as open research questions.

Findings

Generated explicit formulas for tiling counts.

Introduced an alternative computational approach.

Proposed new classes of tilings for future enumeration.

Abstract

This article addresses the problem of enumerating the tilings of a plane by lozenges, under the restriction that these tilings be doubly periodic. Kasteleyn's Pfaffian method is applied to compute the generating function of those permutations. The monomials of this function represent the different types of tilings, grouping them according to the number of lozenges in each orientation. We present an alternative approach to compute these types. Finally, two additional classes of tilings are proposed as open enumeration problems.