Perfect Matchings and the Octahedron Recurrence

Details Perfect Matchings and the Octahedron Recurrence Perfect Matchings and the Octahedron Recurrence

2004-02-27

arxiv.org/abs/math/0402452v2

Mathematics

Combinatorics

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Scientific paper

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky.

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