Existence of the normalized asymptotic limit for higher-dimensional partitions

Prove that the limit of n^{-d/(d+1)} log p_d(n) exists for every dimension d >= 3, where p_d(n) denotes the number of d-dimensional partitions of n.

Background

The paper reviews bounds showing that log p_d(n) has order n{d/(d+1)} and that, for d >= 7, the lower bound exceeds the leading asymptotic coefficient of the MacMahon numbers. These bounds do not establish convergence of the normalized quantity n{-d/(d+1)} log p_d(n).

The numerical analysis assumes an asymptotic expansion for log p_d(n), but the existence of the leading normalized limit remains an unresolved analytic issue for dimensions d >= 3. Consequently, the reported coefficients are numerical estimates rather than rigorous limits.

References

However, note that it is not yet rigorously proven that the limit $_{n \to \infty}n{-\frac{d}{d+1}\log p_d(n)$ exists for $d \geq 3$.

— Estimating the asymptotics of integer partitions in intermediate dimensions ($d = 3,4,5,6$)  (2609.03034 - Mondal, 2 Sep 2026) in Section 2, paragraph following Eq. (Damir-result)

Note that it was conjectured in that $\alpha_r{(d)} = \mu_r{(d)}$, $\nu{(d)} = \beta{(d)}$ and $\epsilon{(d)} = \eta{(d)}$ for all $d$.

— Estimating the asymptotics of integer partitions in intermediate dimensions ($d = 3,4,5,6$)  (2609.03034 - Mondal, 2 Sep 2026) in Section 2, paragraph following the asymptotic-expansion assumption