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

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

2003-09-19

arxiv.org/abs/math/0309332v3

Discrete & Computational Geometry 32 (2004), 437-446 (special issue in honor of Louis Billera)

Mathematics

Combinatorics

9 pages, 2 figures

Scientific paper

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.

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