Enumeration of lozenge tilings of a hexagon with a maximal staircase and a unit triangle removed

TL;DR

This paper derives formulas for counting lozenge tilings of hexagons with specific holes, extending previous results by including a unit triangle removal and multiple holes, using graphical condensation techniques.

Contribution

It introduces new weighted and unweighted formulas for hexagons with a unit triangle removed, generalizing prior tiling enumeration results.

Findings

Derived formulas for tilings with a unit triangle removed

Extended to regions with three holes on consecutive edges

Applied Kuo's graphical condensation and Ciucu's factorization theorem

Abstract

Proctor proved a formula for the number of lozenge tilings of a hexagon with side-lengths $a,b,c,a,b,c$ after removing a "maximal staircase." Ciucu then presented a weighted version of Proctor's result. Here we present weighted and unweighted formulas for a similar region which has an additional unit triangle removed. We use Kuo's graphical condensation method to prove the results. By applying the factorization theorem of Ciucu, we obtain a formula for the number of lozenge tilings of a hexagon with three holes on consecutive edges.