The generating function of lozenge tilings for a "quarter" of a hexagon, obtained with non--intersecting lattice paths

TL;DR

This paper demonstrates how to derive the generating functions for lozenge tilings of quartered hexagons with dents using the Lindström–Gessel–Viennot method, providing an alternative to graphical condensation.

Contribution

It offers a new proof technique for existing results on lozenge tilings using non-intersecting lattice paths and determinant evaluations.

Findings

Derivation of generating functions via Lindström–Gessel–Viennot method

Alternative proof to graphical condensation approach

Extension of tiling enumeration to quartered hexagons with dents

Abstract

In a recent preprint, Lai and Rohatgi compute the generating functions of lozenge tilings of "quartered hexagons with dents" by applying the method of "graphical condensation". The purpose of this note is to exhibit how (a generalization of) Theorems 2.1 and 2.2 in Lai and Rohatgi's preprint can be achieved by the Lindstr\"om--Gessel--Viennot method of non--intersecting lattice paths and a certain determinant evaluation.