AI 中文总结
本文研究矩形分拆函数$p(m,n)$的渐近行为，通过初等方法推导其对数渐近公式，证实相关猜想并推广整数分拆的Hardy-Ramanujan公式。
AI 中文摘要
设$p(m,n)$表示将$m\times n$矩形分拆为整数边长矩形块的分拆数，若两个分拆的块多重集相同则视为不可区分，与几何排列无关。本文提出初等方法证明：对每个固定正整数$m$，当$n\to\infty$时，$\log p(m,n)=\pi\sqrt{\frac{2mH_m}{3}}\sqrt{n}+O(\log n)$，其中$H_m$为第$m$个调和数。该结果证实了作者近期提出的猜想，并推广了整数分拆的Hardy-Ramanujan公式。
英文摘要
Let $p(m,n)$ denote the number of partitions of a rectangle $m\times n$ into integer-sided rectangular blocks, where two partitions are indistinguishable if they consist of the same multiset of blocks, regardless of their geometric arrangement. We present an elementary approach to show that, for every fixed positive integer $m$, $$ \log p(m,n)=π\sqrt{\tfrac{2mH_m}{3}}\sqrt{n}+O(\log n), \qquad \text{as }n\to\infty, $$ where $H_m$ denotes the $m$-th harmonic number. This confirms a conjecture recently posed by the authors and generalizes the Hardy--Ramanujan formula for integer partitions.
Comments10 pages, 1 figure
