The Cube Recurrence
Gabriel D. Carroll, David E Speyer

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 . 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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Topics in Algebra
