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

2020/07/09 by Markus Fulmek, Fulmek, Markus
Mathematics · Physics and Astronomy · Materials Science · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2007.05061

Abstract

In a recent preprint, Lai worked out the quotient of generating functions of\nweighted lozenge tilings of two "half hexagons with lateral dents" which differ\nonly in width. Lai achieved this by using "graphical condensation" (i.e.,\napplication of a certain Pfaffian identity to the weighted enumeration of\nmatchings).\n The purpose of this note is to exhibit how this can be done by the\nLindstr "om--Gessel--Viennot method for nonintersecting lattice paths in a\nquite simple way. Basically the same observation, but restricted to mere\nenumeration (i.e., all weights of lozenge tilings are equal to 1), is\ncontained in a recent preprint of Condon.\n

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