2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68625We present a new tool to compute the number $ϕ_\A (\b)$ of integer solutions to the linear system $$ \x \geq 0 \qquad \A \x = \b $$ where the coefficients of $\A$ and $\b$ are integral. $ϕ_\A (\b)$ is often described as a \emph{vector partition function}. Our methods use partial fraction expansions of Euler's generating function for $ϕ_\A (\b)$. A special class of vector partition functions are Ehrhart (quasi-)polynomials counting integer points in dilated polytopes.9 pages, 2 figuresCombinatorics05A15, 52C07; 52C45text