Trimer covers in the triangular grid: twenty mostly open problems

TL;DR

James Propp introduces the trimer-cover framework on benzels, finite subgraphs of the triangular lattice tiled by five prototiles (right/left stones and vertical/rising/falling bones), and relates trimer tilings to dual tilings via $(a,b)$-benzels. He organizes the problem space into 20 open questions across all prototile-availability regimes, using the Conway–Lagarias invariant to constrain tilings and employing computational enumeration to collect data that suggests exact formulas in many regimes. Several central-diagonal and peripheral-diagonal conjectures are proposed, including exact closed forms for specific $(a,b)$-benzel families, and intriguing $p$-adic phenomena emerge in all-prototile cases, indicating rich arithmetic structure beyond simple counting. The work aims to stimulate exact enumeration results in the spirit of rhombus-tilings and Aztec-diamond domino tilings, with connections to ribbon tilings, Schröder numbers, and royal-path counts, and it highlights numerous open directions for combinatorial and algebraic refinement.

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.

Figures (5)

• Figure 1: The (9,9)-benzel tiled by stones and bones and the associated trimer cover of the (9,9)-benzel graph.
• Figure 2: The (4,6)-benzel and its enclosing hexagon.
• Figure 3: Barycentric coordinates for the cells of the (4,6)-benzel.
• Figure 4: The five prototiles: the right stone, the left stone, the vertical bone, the rising bone, and the falling bone.
• Figure 5: 2-flips for changing a trimer cover.