The quotient of generating functions of lozenge tilings for certain regions derived from hexagons, obtained with non--intersecting lattice paths

TL;DR

This paper demonstrates how to derive the quotient formulas for weighted lozenge tilings of certain hexagon-derived regions using the Lindström–Gessel–Viennot method, providing an alternative to graphical condensation.

Contribution

It introduces a new lattice path approach to obtain quotient formulas for lozenge tilings, complementing existing graphical condensation techniques.

Findings

Quotient of generating functions factors nicely for specific regions.

Lindström–Gessel–Viennot method applied successfully to these tiling problems.

Provides an alternative proof to existing results using nonintersecting lattice paths.

Abstract

In a recent preprint, Lai showed that the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents", which differ only in width, factors nicely, and the same is true for the quotient of generating functions of weighted lozenge tilings of two "quarter hexagons with lateral dents". Lai achieved this by using "graphical condensation" (i.e., application of a certain Pfaffian identity to the weighted enumeration of matchings). The purpose of this note is to exhibit how this can be done by the Lindstr\"om--Gessel--Viennot method for nonintersecting lattice paths. For the case of "half hexagons", basically the same observation, but restricted to mere enumeration (i.e., all weights of lozenge tilings are equal to $1$), is contained in a recent preprint of Condon.