[Paper Review] Another dual of MacMahon's theorem on plane partitions

Mihai Ciucu|arXiv (Cornell University)|Sep 21, 2015

Advanced Combinatorial Mathematics10 references7 citations

TL;DR

This paper introduces the 'fern'—a structure of alternating-oriented equilateral triangles aligned along a lattice line—as a dual counterpart to MacMahon's plane partition theorem. By embedding the fern at the center of a hexagonal region on the triangular lattice, the authors derive a product formula for the normalized number of lozenge tilings of the exterior region, generalizing MacMahon's classical result and revealing a deeper duality between the fern and shamrock structures.

ABSTRACT

In this paper we introduce a counterpart structure to the shamrocks studied in the paper "A dual of Macmahon's theorem on plane partitions" by M. Ciucu and C. Krattenthaler (Proc. Natl. Acad. Sci. USA, vol. 110 (2013), 4518-4523), which, just like the latter, can be included at the center of a lattice hexagon on the triangular lattice so that the region obtained from the hexagon by removing it has its number of lozenge tilings given by a simple product formula. The new structure, called a fern, consists of an arbitrary number of equilateral triangles of alternating orientations lined up along a lattice line. The shamrock and the fern seem to be the only structures with this property. It would be interesting to understand why these are the only two such structures.

Motivation & Objective

• To extend MacMahon's theorem on plane partitions by introducing a new class of lattice structures with simple tiling product formulas.
• To establish a duality between the newly defined 'fern' structure and the previously studied 'shamrock' structure in tiling enumeration.
• To demonstrate that the fern, like the shamrock, allows for a normalized tiling count given by a product formula when embedded in a hexagonal region.
• To investigate why only the fern and shamrock structures exhibit this tiling property, suggesting a fundamental combinatorial uniqueness.

Proposed method

• Define the fern $ F(a_1, \dotsc, a_k) $ as a sequence of equilateral triangles of alternating orientations along a lattice line, with side lengths $ a_1, \dotsc, a_k $.
• Introduce the exterior region $ F^*(a_1, \dotsc, a_k) $, which is the hexagon with the fern removed, and study its lozenge tiling count $ \operatorname{M}(F^*(a_1, \dotsc, a_k)) $.
• Define a normalized ratio of tiling counts between $ F^*(a_1, \dotsc, a_k) $ and $ F^*(o,e) $, where $ o = a_1 + a_3 + \cdots $, $ e = a_2 + a_4 + \cdots $, using a limit as $ N \to \infty $ of tiling counts in $ H_N(a_1, \dotsc, a_k) $.
• Use the Cohn-Larsen-Propp interpretation of the Gelfand-Tsetlin result to express the number of tilings of semihexagonal regions $ S(b_1, \dotsc, b_l) $ via hyperfactorials and alternating sums of $ b_i $.
• Derive a recurrence for tiling functions using a recursive decomposition of the hexagonal regions and verify consistency via algebraic identities.
• Establish a geometric interpretation showing that the normalized tiling ratio equals the number of tilings of the smallest balanced hexagon containing the fern, $ \operatorname{M}(H_F) $.

Experimental results

Research questions

• RQ1 What structures, besides the shamrock, allow for a simple product formula for the number of lozenge tilings of their exterior when embedded in a hexagonal region?
• RQ2 How does the fern structure generalize MacMahon’s classical plane partition formula?
• RQ3 Why do only the fern and shamrock structures exhibit the property of yielding a simple product formula for their exterior tiling count?
• RQ4 Can the normalized tiling ratio of a fern be expressed in terms of known tiling formulas for semihexagonal regions?
• RQ5 Is there a geometric interpretation that explains the product formula for the fern’s exterior tiling count?

Key findings

• The normalized tiling ratio of the fern’s exterior is given by $ \frac{\operatorname{M}(F^*(a_1,\dotsc,a_k))}{\operatorname{M}(F^*(o,e))} = s(a_1,\dotsc,a_{k-1}) \cdot s(a_2,\dotsc,a_k) $, where $ s $ denotes the number of tilings of a semihexagonal region with alternating removed and intact segments.
• When $ k = 3 $, the formula reduces to MacMahon’s original plane partition formula, showing the fern generalizes the classical result.
• The normalized tiling ratio equals $ \operatorname{M}(H_F) $, the number of tilings of the smallest balanced hexagon containing the fern, providing a geometric interpretation.
• The tiling ratio remains invariant under certain parameter shifts, and the limit in the definition of the normalized ratio stabilizes even for finite $ N $, not just in the $ N \to \infty $ limit.
• The recurrence relations for tiling functions are verified algebraically, confirming consistency with the derived product formula.
• The fern and the shamrock are conjectured to be the only two structures with this tiling duality property, suggesting a deep combinatorial uniqueness.