Consider the integer points in a polytope and connect with an edge every pair who are neighbors. This gives the set of edges of a useful simplicial complex called the neighborhood complex. Using the fact that the neighborhood complex is contractible, we can compute generating functions for the following types of sets: those defined as the projection of the set of integer points of a polyhedron.
As a concrete example of this sort of set, suppose we are given relatively prime positive integers a_1, ... ,a_d, and define S to be the set of integers that can be written as a nonnegative integer combination of these a_i. We'd like to answer questions like what is the largest integer not in S and how many integers are not in S. These questions can be attacked via generating functions.
This talk builds on joint work with Herbert Scarf.