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

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