The partial-fractions method for counting solutions to integral linear systems

Abstract

We present a new tool to compute the number φA(b) of integer solutions to the linear system x ≥ 0, Ax = b, where the coefficients of A and b are integral. φA(b) is often described as a vector partition function. Our methods use partial fraction expansions of Euler's generating function for φA(b). A special class of vector partition functions are Ehrhart (quasi-)polynomials counting integer points in dilated polytopes.