Computing vector partition functions

TL;DR

This paper introduces a new algorithm for computing closed-form quasi-polynomial formulas for vector partition functions, implemented in a computer algebra system, with an elementary and self-contained exposition of the underlying theory.

Contribution

The paper presents a novel algorithm for calculating vector partition functions as quasi-polynomials, with an implementation in a computer algebra system and a clear, accessible exposition of the theory.

Findings

Algorithm successfully computes closed-form formulas

Implementation available in the 'calculator' system

Exposition clarifies existing theory

Abstract

A vector partition function is the number of ways to write a vector as a non-negative integer-coefficient sum of the elements of a finite set of vectors $\Delta$. We present a new algorithm for computing closed-form formulas for vector partition functions as quasi-polynomials over a finite set of pointed polyhedral cones, implemented in the ``calculator'' computer algebra system. We include an exposition of previously known theory of vector partition functions. While our results are not new, our exposition is elementary and self-contained.