Conjecture on the asymptotic behavior of the rectangle partition function p(m,n)

Let $p(m,n)$ be the number of partitions of an $m\times n$ rectangle into integer-sided rectangular blocks, where partitions are identified when they consist of the same multiset of blocks, regardless of geometric arrangement. For every fixed positive integer $m$ , as $n\to\infty$ , one conjectures that $\log p(m,n)=\pi\sqrt{\frac{2mH_m}{3}}\sqrt{n}+O(\log n)$ , where $H_m=\sum_{k=1}^{m}\frac{1}{k}$ is the $m$ -th harmonic number.

References

Primary source

arXiv

Additional references

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An August 24, 2026 preprint claims to settle the conjecture in every fixed dimension, but the claim has not been independently checked.

The conjecture predicts the leading growth of rectangle partitions as $n$ tends to infinity with fixed $m$ . The retrieved sources give no proposer or original date.

Known results

$m=1$ : the Hardy–Ramanujan asymptotic.
$m=2$ : $p(2,n)\sim\frac{\pi\sqrt[4]{2}}{32n^{7/4}}\exp(\pi\sqrt{2n})$ .
$m=3$ : $\log p(3,n)=\pi\sqrt{11n/3}\,\sqrt n+O(\log n)$ , determining the conjectured leading constant.

August 24, 2026 claimed proof

On August 24, 2026, The asymptotic behavior of the rectangle partition function claimed, for every fixed positive integer $m$ , $\log p(m,n)=\pi\sqrt{2mH_m/3}\,\sqrt n+O(\log n)$ . This unrefereed preprint conflicts with the earlier statement that cases $m\ge4$ were open; no independent verification or referee assessment was found.

Current status (as of August 2026): The cases $m=1$ , $m=2$ , and $m=3$ are supported by the retrieved literature, while the all- $m$ result, including $m\ge4$ , remains unsettled because the August 24 claim is unverified.

Sources

No solutions have been posted yet.