Bounds for the number of multidimensional partitions

We obtain estimates for the number $p_d(n)$ of $(d-1)$-dimensional integer partitions of a number $n$. It is known that the two-sided inequality $C_1(d)n^{1-1/d}<\log p_d(n)< C_2(d)n^{1-1/d}$ is always true and that $C_1(d)>1$ whenever $\log n> 3d$. However, establishing the $``$right$"$ dependence of $C_2$ on $d$ remained an open problem. We show that if $d$ is sufficiently small with respect to $n$, then $C_2$ does not depend on $d$, which means that $\log p_d(n)$ is up to an absolute constant equal to $n^{1-1/d}$. Besides, we provide estimates of $p_d(n)$ for different ranges of $d$ in terms of $n$, which give the asymptotics of $\log p_d(n)$ in each case.