Computing a pyramid partition generating function with dimer shuffling

TL;DR

This paper verifies a conjecture by computing the generating function for pyramid partitions, linking combinatorial models to Donaldson-Thomas theory without using algebraic geometry, via a modified domino shuffling algorithm.

Contribution

It introduces a combinatorial method to compute the pyramid partition generating function, confirming a conjecture and connecting to non-commutative Donaldson-Thomas theory.

Findings

Confirmed the conjecture of Kenyon/Szendroi

Linked pyramid partitions to Donaldson-Thomas theory of the conifold

Developed a modified domino shuffling algorithm

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.