  Given a_1,a_2,...,a_n in Z^d, we examine the set, G, of all non-negative
integer combinations of these a_i. In particular, we examine the generating
function f(z)=\sum_{b\in G} z^b. We prove that one can write this generating
function as a rational function using the neighborhood complex (sometimes
called the complex of maximal lattice-free bodies or the Scarf complex) on a
particular lattice in Z^n. In the generic case, this follows from algebraic
results of D. Bayer and B. Sturmfels. Here we prove it geometrically in all
cases, and we examine a generalization involving the neighborhood complex on an
arbitrary lattice.
