The Cube Recurrence

TL;DR

This paper introduces a combinatorial model based on the cube recurrence that generates Laurent polynomials, proves related conjectures, and links to Gale-Robinson sequences and perfect matchings in bipartite planar graphs.

Contribution

It constructs a novel combinatorial model for the cube recurrence, proving several conjectures and providing new interpretations for Gale-Robinson sequences.

Findings

Proved several conjectures of Propp, Fomin, and Zelevinsky.

Provided a combinatorial interpretation for Gale-Robinson sequences.

Suggested applications to perfect matchings in bipartite planar graphs.

Abstract

We construct a combinatorial model that is described by the cube recurrence, a nonlinear recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in $\mathbb{Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.