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The Partial-Fractions Method for Counting Solutions to Integral Linear Systems

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Abstract

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

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Correspondence to Matthias Beck.

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Beck, M. The Partial-Fractions Method for Counting Solutions to Integral Linear Systems. Discrete Comput Geom 32, 437–446 (2004). https://doi.org/10.1007/s00454-004-1131-5

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  • Vector partition function
  • Ehrhart theory
  • Littlewood-Richardson
  • Kostant partition function
  • BZ-triangles