Characteristic quasi-polynomials of integral hyperplane arrangements

A typical problem in enumerative combinatorics is to count the size of a set depending upon a positive integer q. Often the result is a polynomial in q (e.g., the chromatic polynomial of a graph), and sometimes a quasi-polynomial. Generally speaking, a quasipolynomial is a generalization of polynomials, of which the coefficients may not come from a ring but instead are periodic functions with integral periods. Another way to think of a quasi-polynomial is that it is made of a bunch of polynomials, called the constituents.