The cube recurrence

Abstract

We construct a combinatorial model that is described by the cube recurrence, a quadratic recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in ℤ3. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky about the structure of these polynomials, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences, including the Somos-6 and Somos-7 sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.