Trimer covers in the triangular grid: twenty mostly open problems

TL;DR

This paper explores the combinatorial problem of trimer covers on triangular grids, introducing new geometric regions called benzels, and discusses numerous open problems and conjectures related to counting these tilings.

Contribution

It introduces benzels as a new class of regions for trimer tilings and highlights twenty open problems, expanding the study beyond traditional dimer tilings.

Findings

Introduction of benzels as a new tiling region

Identification of twenty open problems in trimer tilings

Discussion of conjectural formulas for counting tilings

Abstract

In the past three decades, the study of rhombus tilings and domino tilings of various plane regions has been a thriving subfield of enumerative combinatorics. Physicists classify such work as the study of dimer covers of finite graphs. In this article we move beyond dimer covers to trimer covers, introducing plane regions called benzels that play a role analogous to hexagons for rhombus tilings and Aztec diamonds for domino tilings, inasmuch as one finds many (so far mostly conjectural) exact formulas governing the number of tilings.