[Paper Review] A $q$-enumeration of lozenge tilings of a hexagon with three dents

Tri Lai|arXiv (Cornell University)|Feb 20, 2015

Advanced Combinatorial Mathematics20 references4 citations

TL;DR

This paper presents a $q$-enumeration formula for lozenge tilings of a hexagon with three bowtie-shaped dents on non-consecutive sides, generalizing Propp's problem on central triangle removals. Using $q$-analogs of hyperfactorials and Kuo condensation, the authors derive a closed-form product formula that extends MacMahon's classical plane partition enumeration to defective hexagonal regions with arbitrary dent sizes.

ABSTRACT

We $q$-enumerate lozenge tilings of a hexagon with three bowtie-shaped regions have been removed from three non-consecutive sides. The unweighted version of the result generalizes a problem posed by James Propp on enumeration of lozenge tilings of a hexagon of side-lengths $2n,2n+3,2n,2n+3,2n,2n+3$ (in cyclic order) with the central unit triangles on the $(2n+3)$-sides removed.

Motivation & Objective

• To generalize James Propp's 1999 problem on lozenge tilings of a hexagon with three central unit triangles removed to arbitrary dent sizes.
• To extend the $q$-enumeration framework to hexagonal regions with three non-consecutive bowtie-shaped dents (each consisting of two adjacent triangles).
• To derive a closed-form product formula for the $q$-enumeration of lozenge tilings in such defective hexagonal regions.
• To establish a connection between these tiling enumerations and constrained plane partitions with specific boundary conditions.
• To demonstrate the power of Kuo condensation in proving $q$-enumeration identities in tiling theory.

Proposed method

• Define the region $F\begin{pmatrix}x&y&z\\ a&b&c\\ d&e&f\end{pmatrix}$ as a hexagon with three bowtie-shaped dents on non-consecutive sides, parameterized by nine non-negative integers.
• Apply $q$-analogues of hyperfactorials ($\operatorname{H}_q(n)$) and $q$-integers ($[n]_q$) to encode the weight of each tiling by the volume of the corresponding plane partition.
• Use Kuo condensation as the primary combinatorial technique to derive recurrence relations for the $q$-enumeration function $\Psi$.
• Establish functional equations for the $q$-enumeration by comparing ratios of $\Psi$ values under parameter shifts, leading to a system of identities.
• Prove the consistency of the recurrence by verifying a key identity involving $q$-integers: $\frac{[a+d+x+y]_q}{[a+c+d+f+2x+y+z]_q} + \frac{q^{a+d+x+y}[c+f+x+z]_q}{[a+c+d+f+2x+y+z]_q} = 1$.
• Leverage the uniqueness of the solution to the recurrence to conclude the closed-form formula for the $q$-enumeration.

Experimental results

Research questions

• RQ1 What is the $q$-enumeration of lozenge tilings in a hexagon with three bowtie-shaped dents on non-consecutive sides?
• RQ2 How does the $q$-enumeration generalize the unweighted count of lozenge tilings in Propp's original problem?
• RQ3 Can Kuo condensation be systematically applied to derive $q$-enumeration formulas for defective hexagonal regions?
• RQ4 What constraints do forced lozenges impose on the corresponding plane partitions, and how can these be encoded algebraically?
• RQ5 What is the exact closed-form expression for the $q$-enumeration of lozenge tilings in the generalized region with arbitrary dent sizes?

Key findings

• The $q$-enumeration of lozenge tilings of the region $F\begin{pmatrix}x&y&z\\ a&b&c\\ d&e&f\end{pmatrix}$ is given by a product formula involving $q$-hyperfactorials $\operatorname{H}_q(n)$, with explicit dependence on all nine parameters.
• The formula reduces to MacMahon's classical $q$-enumeration of plane partitions in the case $x=y=z=a=b=c=d=e=f=0$, recovering the standard hexagon tiling count.
• When $d=e=f=0$, the region $N_{a,b,c}(x,y,z)$ corresponds to a hexagon with three triangular dents, and the $q$-enumeration yields a formula for plane partitions with specific boundary constraints.
• The derived formula satisfies a recurrence derived via Kuo condensation, and the consistency of this recurrence is verified using a fundamental identity in $q$-integers.
• The result generalizes Eisenkölbl's solution to Propp's problem by allowing arbitrary dent sizes instead of just unit triangles.
• The method provides a systematic approach to proving other $q$-enumeration identities in tiling theory, suggesting broader applicability of Kuo condensation in $q$-combinatorics.