Perfect matchings and the octahedron recurrence
Journal of Algebraic Combinatorics, Tome 25 (2007) no. 3, pp. 309-348.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Keywords: keywords aztec diamond, perfect matching, octahedron recurrence, somos sequence, somos four, somos five, cluster algebra
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Speyer, David E. Perfect matchings and the octahedron recurrence. Journal of Algebraic Combinatorics, Tome 25 (2007) no. 3, pp. 309-348. http://geodesic.mathdoc.fr/item/JAC_2007__25_3_a1/