Computing a pyramid partition generating function with dimer shuffling

TL;DR

The paper addresses computing the pyramid-partition generating function $Z_A^{(n)}(q_0,-q_1)$ and verifies Kenyon/Szendroi's conjecture by mapping pyramid partitions to dimer covers and applying a weight-preserving, modified EKLP domino shuffling. It develops a length-$\infty$ analysis via a weight-preserving bijection to (super-)rigid 3D partitions, enabling explicit generating functions and linking to Donaldson–Thomas theory of a conifold resolution. For $n=1$, it derives a product formula and extends to general $n$ by topological-vertex evaluations, yielding a closed form in terms of MacMahon functions and infinite products. The methods illuminate connections between pyramid partitions, dimer models, and DT invariants, and suggest refinements and geometry-generalizations to broader singularities and multi-leg configurations.

Abstract

We verify a recent conjecture of Kenyon/Szendroi, arXiv:0705.3419, by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson--Thomas theory of a non-commutative resolution of the conifold singularity {x1x2 -x3x4 = 0}. The proof does not require algebraic geometry; it uses a modified version of the domino shuffling algorithm of Elkies, Kuperberg, Larsen and Propp.

Figures (13)

• Figure 1: Special bricks, assembled into the configuration $\varepsilon_3$.
• Figure 2: A pyramid partition of length 1, viewed from the side and from above
• Figure 3: The empty rooms of lengths 1 and 2.
• Figure 4:
• Figure 5:

Theorems & Definitions (27)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof