As an example, consider the following problem. Given positive integers $a_1,…,a_d$ that are relatively prime, let S be the set of integers that can be written as a nonnegative integer combination of these $a_i$. We can think of the $a_i$ as denominations of postage stamps and S as the postal rates that can be paid exactly using these denominations. What can we say about the structure of this set, S? What is the largest integer not in S (called the Frobenius number)? How many positive integers are not in S? We attack these problems using the generating function $f_S(x)$, defined to be the sum, over all elements s of S, of the monomials $x^s$. We will build up the general theory of computing generating functions – for this and other problems – and then use these generating functions to answer questions we’re interested in. We will approach these problems from an algorithmic perspective: what can we do in polynomial time?