Lozenge Tilings of Hexagons with Intrusions I: Generalized Intrusion

TL;DR

This paper extends classical lozenge tiling enumeration of hexagons by introducing intrusions, providing new product formulas and q-analogues for the number of tilings of these generalized regions.

Contribution

It introduces a new class of regions with intrusions in hexagons and derives simple product formulas and q-analogues for their lozenge tiling counts.

Findings

Derived explicit product formulas for tilings of intruded hexagons.

Established q-analogues for the enumeration of these tilings.

Extended classical results to more complex regions with holes.

Abstract

MacMahon's classical theorem on the number of boxed plane partitions has been generalized in several directions. One way to generalize the theorem is to view boxed plane partitions as lozenge tilings of a hexagonal region and then generalize it by making some holes in the region and counting its tilings. In this paper, we provide new regions whose numbers of lozenges tilings are given by simple product formulas. The regions we consider can be obtained from hexagons by removing structures called intrusions. In fact, we show that the tiling generating functions of those regions under certain weights are given by similar formulas. These give the $q$-analogue of the enumeration results.