A certain ratio of generating functions of lozenge tilings, obtained with non--intersecting lattice paths

TL;DR

This paper demonstrates a simplified method using nonintersecting lattice paths to derive ratios of generating functions for lozenge tilings, building on prior graphical condensation techniques.

Contribution

It introduces a straightforward approach employing Lindström–Gessel–Viennot paths to obtain generating function ratios, simplifying previous Pfaffian-based methods.

Findings

Simplified derivation of generating function ratios

Connection between graphical condensation and lattice path methods

Extension to enumeration of unweighted lozenge tilings

Abstract

In a recent preprint, Lai worked out the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents" which differ only in width. 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 in a quite simple way. 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.