Lozenge Tilings of Hexagons with Intrusions II: Shuffling Phenomenon

TL;DR

This paper studies lozenge tilings of hexagons with holes, introducing new hybrid regions and shuffling theorems that yield simple product formulas for their tiling generating functions.

Contribution

It introduces new hybrid regions combining previous models and proves shuffling theorems that relate their tiling generating functions with simple formulas.

Findings

Tiling generating functions are given by simple product formulas.

Shuffling theorems relate tilings of related hexagonal regions with intrusions.

New hybrid regions extend previous models of hexagon tilings.

Abstract

The enumeration of lozenge tilings of hexagons with holes has been studied intensively in recent years. Researchers tried to find shapes and positions of holes in hexagonal regions so that the number of lozenge tilings of the resulting regions is given by a simple product formula. In the present work, we consider new regions that are hybrids of regions studied by the first author (hexagons with intrusions) and Ciucu (F-cored hexagons). Then, we show that the tiling generating functions of these new regions under a certain weight are given by simple product formulas. To give a proof, we present shuffling theorems for lozenge tilings of hexagons with intrusions, which give simple relations between the tiling generating functions of two related hexagonal regions with intrusions.