Cyclically Symmetric Lozenge Tilings of a Hexagon with Four Holes

TL;DR

This paper generalizes the enumeration of cyclically symmetric lozenge tilings of a hexagon by including four central holes, extending previous product formulas for symmetric tilings.

Contribution

It introduces a new enumeration formula for cyclically symmetric lozenge tilings of a hexagon with four central holes, expanding prior results on symmetric plane partitions.

Findings

Derived a product formula for tilings with four holes

Extended symmetry enumeration to more complex hexagon configurations

Generalized previous results on symmetric tilings

Abstract

The work of Mills, Robbins, and Rumsey on cyclically symmetric plane partitions yields a simple product formula for the number of lozenge tilings of a regular hexagon, which are invariant under roation by $120^{\circ}$. In this paper we generalize this result by enumerating the cyclically symmetric lozenge tilings of a hexagon in which four triangles have been removed in the center.