Taken from HAKMEM 18.
Quoting...
A partition of $N$ is a finite string of non-increasing integers that add up to $N$. Thus 7 3 3 2 1 1 1 is a partition of 18. Sometimes an infinite string of zeros is extended to the right, filling a half-line. The number of partitions of $N$, $P(N)$, is a fairly well understood function.
The generating function is:$\sum_{n=0}^{\infty} \; P(n) \; x^n = \dfrac1{\prod\limits_{k=1}^{\infty} (1-x^k)}$
A planar partition is like a partition, but the entries are in a two-dimensional array (the first quadrant) instead of a string. Entries must be non-increasing in both the x and y directions. A planar partition of 34 would be:
$\begin{matrix}1&&&&\\3&1&&&\\3&2&2&1&\\7&6&4&3&1\end{matrix}$
Zeros fill out the unused portion of the quadrant. The number of planar partitions of $n$, $PL(n)$, is not a very well understood function.
The generating function is:$\sum_{n=0}^{\infty} \; PL(n) \; x^n = \dfrac1{\prod\limits_{k=1}^{\infty} (1-x^k)^k}$
Similarly, one can define cubic partitions with entries in the first octant, but no one has been able to discover the generating function. Some counts for cubic partitions and a discussion appear in Knuth, Math. Comp. 1970 or so.
Did anyone find a generating function for cubic partitions since then?