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

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