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arXiv · 1901.00554

Frobenius Coin-Exchange Generating Functions

Leonardo Bardomero·Matthias Beck
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Abstract

We study variants of the \emph{Frobenius coin-exchange problem}: given $n$ positive relatively prime parameters, what is the largest integer that cannot be represented as a nonnegative integral linear combination of the given integers? This problem and its siblings can be understood through generating functions with 0/1 coefficients according to whether or not an integer is representable. In the 2-parameter case, this generating function has an elegant closed form, from which many corollaries follow, including a formula for the Frobenius problem. We establish a similar closed form for the generating function indicating all integers with exactly $k$ representations, with similar wide-ranging corollaries.

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Leonardo Bardomero, Matthias Beck. 2019-01-02. Frobenius Coin-Exchange Generating Functions. https://arxiv.org/abs/1901.00554

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