2020/12/13 by Markus Fulmek, Fulmek, Markus
Computer Science · Mathematics · Physics and Astronomy · #05A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems
paper · pdf · doi:10.48550/arxiv.2012.07168
openalex publication_date 2020/12/13 · openalex created_date 2020/12/21 · openalex updated_date 2026/07/28
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 Lindström--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.