Determine the polynomial prefactor for rectangle partition asymptotics

Determine the polynomial factor multiplying the leading exponential term in the asymptotic formula for the rectangle partition function p(m,n) for each fixed integer m≥4.

Background

The paper establishes, for every fixed positive integer m, the logarithmic asymptotic behavior log p(m,n)=π√((2mH_m/3)√n)+O(log n) as n tends to infinity, where H_m is the m-th harmonic number. For m≤3, earlier work provides sharper asymptotic formulae that include a polynomial factor multiplying the exponential term.

The authors explicitly note that determining this polynomial factor remains unresolved when m≥4. Thus, the open problem concerns refining the established logarithmic estimate into a full asymptotic formula, including the polynomial prefactor, for every fixed m≥4.

References

For $m\le3$, the known asymptotic formulae for $p(m,n)$ exhibit a polynomial factor in front of the exponential term; see . Determining this factor for a fixed $m\geq4$ remains an open problem.

— The asymptotic behavior of the rectangle partition function $p(m,n)$  (2608.22955 - Gajdzica et al., 24 Aug 2026) in Remark following Figure 1, after the proof of Theorem 1; page number unavailable