Perfect Matchings for the Three-Term Gale-Robinson Sequences

In 1991, David Gale and Raphael Robinson, building on explorations carried out by Michael Somos in the 1980s, introduced a three-parameter family of rational recurrence relations, each of which (with suitable initial conditions) appeared to give rise to a sequence of integers, even though a priori the recurrence might produce non-integral rational numbers. Throughout the ’90s, proofs of integrality were known only for individual special cases. In the early ’00s, Sergey Fomin and Andrei Zelevinsky proved Gale and Robinson’s integrality conjecture. They actually proved much more, and in particular, that certain bivariate rational functions that generalize Gale-Robinson numbers are actually polynomials with integer coefficients. However, their proof did not offer any enumerative interpretation of the GaleRobinson numbers/polynomials. Here we provide such an interpretation in the setting of perfect matchings of graphs, which makes integrality/polynomiality obvious. Moreover, this interpretation implies that the coefficients of the Gale-Robinson polynomials are positive, as Fomin and Zelevinsky conjectured. In memory of David Gale, 1921-2008