Tilted Halved Hexagons: Hexagons, Semi-hexagons, and Halved Hexagons Under One Roof

TL;DR

This paper introduces a new family of tiling regions that generalize well-known shapes, providing a simple product formula for their tilings and connecting to enumerations of restricted plane partitions.

Contribution

It presents a universal generalization of three classical tiling regions and derives a simple product formula for their enumeration, linking to restricted plane partitions.

Findings

Derived a product formula for tilings of the new regions

Unified enumeration of hexagons, semi-hexagons, and halved hexagons

Connected tiling enumeration to restricted plane partitions

Abstract

We investigate a new family of regions that is the universal generalization of three well-known region families in the field of enumeration of tilings: the quasi-regular hexagons, the semi-hexagons, and the halved hexagons. We prove a simple product formula for the number of tilings of these new regions. Our main result also yields the enumerations of two special classes of plane partitions with restricted parts.