[Paper Review] Lozenge tilings of doubly-intruded hexagons

Mihai Ciucu, Tri Lai|arXiv (Cornell University)|Dec 21, 2017

Advanced Combinatorial Mathematics19 references3 citations

TL;DR

This paper presents explicit product formulas for the number of lozenge tilings of hexagonal regions with two symmetrically placed fern-like intrusions (removals of contiguous triangular holes), generalizing MacMahon's classical formula. Using Kuo condensation and $q$-enumeration techniques, the authors derive closed-form expressions with linear factors in parameters, contrasting with the complex prime factorizations seen in intruded Aztec diamonds, and pose statistical mechanics questions on tiling limits and phase transitions in such regions.

ABSTRACT

Motivated in part by Propp's intruded Aztec diamond regions, we consider hexagonal regions out of which two horizontal chains of triangular holes (called ferns) are removed, so that the chains are at the same height, and are attached to the boundary. By contrast with the intruded Aztec diamonds (whose number of domino tilings contain some large prime factors in their factorization), the number of lozenge tilings of our doubly-intruded hexagons turns out to be given by simple product formulas in which all factors are linear in the parameters. We present in fact $q$-versions of these formulas, which enumerate the corresponding plane-partitions-like structures by their volume. We also pose some natural statistical mechanics questions suggested by our set-up, which should be possible to tackle using our formulas.

Motivation & Objective

• To generalize MacMahon's lozenge tiling formula for regular hexagons to regions with two symmetrically placed fern-like intrusions.
• To derive simple, closed-form product formulas for the number of lozenge tilings in such doubly-intruded hexagons.
• To extend these formulas to $q$-enumerations that count tilings by volume, providing a generating function framework.
• To explore statistical mechanics questions about tiling behavior under large-scale limits, particularly the effect of intrusions on limit shapes and phase transitions.

Proposed method

• Application of Kuo condensation to derive recurrence relations for tiling counts in hexagonal regions with fern-shaped holes.
• Use of hyperfactorials functions $\operatorname{H}(n)$ to express tiling counts in terms of alternating products over partial sums of lobe sizes.
• Construction of $q$-analogues of the tiling formulas to enumerate tilings by their volume, using Schur functions and symmetric function identities.
• Identification of tiling configurations via forced lozenge arguments and lattice path interpretations, particularly in the unweighted case.
• Use of the Cohn-Larsen-Propp interpretation of Gelfand-Tsetlin determinants to relate tiling counts to Schur functions.
• Derivation of a key Schur function identity (Equation 9.5) that connects tiling enumeration to symmetric function theory.

Experimental results

Research questions

• RQ1 What is the exact number of lozenge tilings of a hexagon with two symmetrically placed ferns removed along a common horizontal line?
• RQ2 Can the tiling count be expressed as a simple product formula with linear factors in the parameters, as in MacMahon’s original formula?
• RQ3 How do $q$-enumerations of these tilings behave, and what is their connection to Schur functions and symmetric function identities?
• RQ4 What is the asymptotic behavior of the tiling count as the region size grows, particularly in relation to limit shapes and phase transitions?
• RQ5 At what height and ratio of slit lengths does the number of tilings achieve a maximum in the large-scale limit?

Key findings

• The number of lozenge tilings of a doubly-intruded hexagon with ferns of sizes $a_1, \dots, a_m$ and $b_1, \dots, b_n$ is given by a product formula involving hyperfactorials functions of partial sums of the lobe sizes.
• The $q$-enumeration of tilings is expressed as a ratio of products of hyperfactorials functions and Schur functions, yielding a rational generating function in $q$.
• The tiling count remains a product of linear terms in parameters, in contrast to intruded Aztec diamonds whose tiling counts contain large prime factors.
• A Schur function identity (Equation 9.5) is derived that relates tiling enumeration to symmetric function theory, with explicit evaluation in terms of hyperfactorials.
• In the asymptotic limit, the maximum number of tilings occurs when the two ferns have equal length ($r_p = 1$), suggesting symmetry maximizes tiling entropy.
• The authors conjecture that for fixed $p = M/(M/2 + N)$, the maximum tiling count occurs when the two intrusions are symmetric, i.e., $r_p = 1$, and pose the open problem of determining the optimal height $h_p$.