4 Perfect Matchings and The Octahedron Recurrence

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. 1 Historical Introduction The octahedron recurrence is the product of three chains of research seeking a common generalization. The first is the study of the algebraic relations between the connected minors of a matrix, and particularly of a recurrence relating them known and Dodgson condensation. Attempting to understand the combinatorics of Dodgson condensation lead to the discovery of alternating sign matrices and Aztec diamonds. Aztec diamonds are graphs whose perfect matchings have extremely structured combinatorics and soon formed their own, second line of research as other graphs were discovered with the similar regularities. The third is the study of Somos-sequences and the Laurent phenomenon, which began with an attempt to understand theta functions from a combinatorial perspective. In this section we will sketch both lines of research. In the following section, we will begin to describe the vocabulary and main results of this paper.