Tiling generating functions of halved hexagons and quartered hexagons

Abstract

We prove exact product formulas for the tiling generating functions of various halved hexagons and quartered hexagons with defects on boundary. Our results generalize the previous work of the first author and the work of Ciucu.

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

“…Theorems 2.1 and 2.2 in [7] state that this determinant factorizes completely for k = 0 and k = 1: Here, we shall show that this is true for general k ≥ 0.…”

“…Lai and Rohatgi [7, Theorems 2.1-2.4] computed the generating functions of lozenge tilings for such "quarter hexagons with dents" of odd or even heights, and with labels starting at 0 or 1 (see Figures 1 and 2), which resulted in four theorems: The case where the labelling starts at 0 (see Figure 2) requires a slight modification of the weight function, but we shall only consider the other case (labelling starts at 1, see Figure 1) and give a generalization which contains the cases of odd and even heights, thus giving an alternative proof for Theorems 2.1 and 2.2 in [7].…”

“…In a recent preprint, Lai and Rohatgi [7] compute the generating functions of lozenge tilings of "quartered hexagons with dents"; their method of proof is induction based on "graphical condensation" (i.e., an application of a certain Pfaffian identity to the enumeration of matchings).…”

“…The purpose of this note is to exhibit how (a generalization of) Theorems 2.1 and 2.2 in [7] can be achieved by the Lindström-Gessel-Viennot method [8,4] of non-intersecting lattice paths and the evaluation of the corresponding determinant. (Theorems 2.3 and 2.4 in [7] most probably can be achieved in the same way, but we want to be brief here.)…”

“…The purpose of this note is to exhibit how (a generalization of) Theorems 2.1 and 2.2 in [7] can be achieved by the Lindström-Gessel-Viennot method [8,4] of non-intersecting lattice paths and the evaluation of the corresponding determinant. (Theorems 2.3 and 2.4 in [7] most probably can be achieved in the same way, but we want to be brief here.) This note is organized as follows: In section 2, we shall describe the generating functions considered by Lai and Rohatgi, and show how the computation of these generating functions boils down (by the well-known bijective correspondence between lozenge tilings and non-intersecting lattice paths and the well-known Lindström-Gessel-Viennot argument) to the evaluation of a certain determinant.…”