2 problems

Let $p_d(n)$ denote the number of $d$-dimensional partitions of $n$. For solid partitions, $d=4$, and the relevant normalization is $n^{1-1/d}$. MacMahon's numbers are the numbers…

For $d>3$, let $p_d(n)$ denote the number of $d$-dimensional partitions of $n$, with $p_d(0)=1$. A generating function for these numbers is a formal power series whose coefficient…