Frobenius硬币兑换Generating Functions
Frobenius Coin-Exchange Generating Functions
摘要：
我们研究变种的\emph{Frobenius coin-exchange problem}：鉴于$n$积极互素参数，什么是最大的整数，不能表示为一个非负整数线性组合的给定整数？这个问题和它的兄弟可以通过根据整数是否可表示来生成具有0/1系数的函数来理解。在2参数的情况下，这个生成函数有一个优雅的封闭形式，从它可以得出许多推论，包括弗罗贝纽斯问题的公式。我们建立了一个类似的封闭形式的生成函数，表明所有的整数正好$k$表示，具有类似的广泛的推论。
英文摘要：
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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