论文标题
通过限制分区功能，广义的Frobenius问题
The generalized Frobenius problem via restricted partition functions
论文作者
论文摘要
给定相对主要的正整数，$ a_1，\ ldots，a_n $，frobenius编号是最大的整数，没有$ a_1x_1+\ cdots+cdots+a_nx_n $的表示，而非阴性整数$ x_i $。该古典价值最近已被概括：鉴于非负整数$ k $，最大的整数最多是$ k $此类表示形式？其他经典价值也可以推广：例如，最多可在$ k $的方式中代表多少个非负整数？对于足够大的$ k $，我们通过了解限制分区功能的级别集（功能$ f（t）$，给出$ t $的表示数）来为这些值提供公式。此外，我们给出了所有这些价值观的完整渐近学，并为某些特殊情况（例如$ n = 2 $案例和文献中的某个极端家庭）谴责公式。最后，我们获得了限制分区函数的前两个主项，作为所谓的准多项式。
Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been generalized: given a nonnegative integer $k$, what is the largest integer with at most $k$ such representations? Other classical values can be generalized too: for example, how many nonnegative integers are representable in at most $k$ ways? For sufficiently large $k$, we give formulas for these values by understanding the level sets of the restricted partition function (the function $f(t)$ giving the number of representations of $t$). Furthermore, we give the full asymptotics of all of these values, as well as reprove formulas for some special cases (such as the $n=2$ case and a certain extremal family from the literature). Finally, we obtain the first two leading terms of the restricted partition function as a so-called quasi-polynomial.
