[Paper Review] Enumeration of lozenge tilings of a hexagon with shamrock hole on boundary

Tri Lai|arXiv (Cornell University)|Feb 5, 2015

Advanced Combinatorial Mathematics3 citations

TL;DR

This paper extends Ciucu and Krattenthaler's work on lozenge tilings by studying hexagons with a shamrock-shaped hole on the boundary rather than at the center. Using combinatorial methods and $q$-enumeration techniques, the authors derive a simple product formula for the number of lozenge tilings in these new regions, generalizing MacMahon’s plane partition theorem to boundary holes.

ABSTRACT

Ciucu and Krattenthaler proved a dual of MacMahon's classical theorem on plane partitions by enumerating lozenge tilings of a hexagon with a shamrock hole at the center (Proc. Natl. Acad. Sci. USA, 2013). We consider a new situation when a similar hole appears on the boundary of a hexagon. We prove that the lozenge tilings of new regions are always enumerated by a simple product formula. In addition, we investigate a related problem on $q$-enumeration plane partitions fitting in a compound box.

Motivation & Objective

• To investigate lozenge tilings of hexagonal regions with a shamrock hole located on the boundary, rather than at the center as in prior work.
• To determine whether such tiling regions still admit a closed-form product formula for their enumeration.
• To generalize MacMahon’s classical plane partition theorem to cases involving boundary holes using combinatorial and $q$-enumerative techniques.
• To explore connections between tiling enumeration and $q$-series identities in the context of compound box-shaped plane partitions.

Proposed method

• The authors define a new family of hexagonal regions with a shamrock hole on the boundary, preserving symmetry and tiling feasibility.
• They apply a variant of the Lindström–Gessel–Viennot lemma to count non-intersecting lattice paths corresponding to lozenge tilings.
• A key step involves transforming the tiling enumeration problem into a determinant computation using a weighted lattice path model.
• They derive a product formula for the number of lozenge tilings by analyzing the structure of the region and exploiting symmetry and recursive decomposition.
• The $q$-enumeration of plane partitions fitting in a compound box is analyzed using generating functions and $q$-series identities.
• The method relies on a generalization of the classical MacMahon formula to boundary-located singularities.

Experimental results

Research questions

• RQ1 Does the presence of a shamrock hole on the boundary of a hexagon still allow for a simple product formula in the enumeration of lozenge tilings?
• RQ2 How does the location of the hole—on the boundary versus at the center—affect the tiling count and the underlying combinatorial structure?
• RQ3 Can the $q$-enumeration of plane partitions in a compound box be related to tiling enumeration in regions with boundary holes?
• RQ4 What symmetries or recursive structures emerge in these new tiling regions that support a closed-form solution?
• RQ5 Are there $q$-analogues of the product formula that generalize the classical case?

Key findings

• The number of lozenge tilings of a hexagon with a shamrock hole on the boundary is given by a simple product formula, analogous to MacMahon’s theorem.
• The product formula depends only on the size and position of the hole relative to the boundary, and not on internal structure beyond that.
• The $q$-enumeration of plane partitions fitting in a compound box is shown to be expressible via a determinant formula involving $q$-integers.
• The tiling count remains an integer product formula even when the hole is not centrally located, indicating robustness of the combinatorial structure.
• The derived formula generalizes the central hole result of Ciucu and Krattenthaler to boundary-located singularities.
• The method successfully extends classical tiling enumeration to new geometric configurations with non-central defects.