Lozenge Tilings Of the Equilateral Triangle

TL;DR

This paper investigates the enumeration of incomplete lozenge tilings of an equilateral triangle subdivided into smaller triangles, providing numerical counts for small edge lengths and analyzing the distribution of lozenges.

Contribution

It introduces a method to count and analyze incomplete lozenge tilings of subdivided equilateral triangles, including detailed enumeration for small cases.

Findings

Numerical counts of tilings for n <= 6

Distribution patterns of lozenges in tilings

Refinement of tiling configurations based on lozenge count

Abstract

We consider incomplete tilings of the equilateral triangle of edge length n that is subdivided into n^2 regular equilateral smaller unit triangles. Pairs of the unit triangles that share a side may be converted into lozenges, leaving some subset of the unit triangles untouched. We count numerically these coverings by lozenges and unit triangles for edge lengths n <= 6: the total and the detailed refinement as a function of the number of lozenges.