AI 中文总结
针对几乎有界分拆数问题，在Xin和Zhang2025年算法基础上结合Barvinok算法，提出可处理含固定数量无界元素序列的多项式时间算法，拓展了分拆数计算的适用场景。
AI 中文摘要
西尔维斯特分拆数$d(t; \boldsymbol{A})$用于计数$\sum_{i=1}^{N} a_i x_i = t$的非负整数解个数，其中$\boldsymbol{A}=(a_1,\dots,a_N)$是满足$\gcd(\boldsymbol{A})=1$的正整数序列。2025年，Xin和Zhang给出了当$\boldsymbol{A}$的元素被常数界定时，以$N$为参数的多项式时间算法计算$d(t; \boldsymbol{A})$。本文通过结合Barvinok算法扩展该算法，使其能处理$\boldsymbol{A}$中有固定数量元素无界的情况。
英文摘要
Sylvester's denumerant $d(t; \boldsymbol{A})$ counts the number of nonnegative integer solutions to $\sum_{i=1}^{N} a_i x_i = t$, where $\boldsymbol{A} = (a_1, \dots, a_N)$ is a sequence of positive integers with $\gcd(\boldsymbol{A}) = 1$. In 2025, Xin and Zhang gave a polynomial time algorithm in $N$ for computing $d(t; \boldsymbol{A})$ when the entries of $\boldsymbol{A}$ are bounded by a constant. In this paper, we extend this algorithm by incorporating Barvinok's algorithm, enabling it to handle the case where a fixed number of entries of $\boldsymbol{A}$ are allowed to be unbounded.
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