The asymptotic behavior of the rectangle partition function $p(m,n)$

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 $m$, $$\log p(m,n)=\pi \sqrt{\frac{2m{H}_m}{3}}\sqrt{n}+O(\log n),\text{as }n\to \mathrm{\infty },$$ 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.