Fix a prime $p$, and $k$ a natural number. The question is then:
How many partitions of $p^k$ are there into powers of $p$?
So, for instance, if $p = 2$ and $k = 2$, there are 4, namely (4), (2, 2), (2, 1, 1) and (1, 1, 1, 1).
This seems like it could be a hard problem in number theory, but I don't really know much number theory (I'm a topologist), and so don't have a feel for such things.
My question 428904, "how many ways to make change, asymptotics" asked the same question and I found the answer in a 1948 paper in Indagationes Math X "On Mahler's Partition Problem". NG De Bruijn cited Karl Mahler's 1940 paper from Journ. London Math Soc. "On a special functional equation". His formula, for base $p=r$ and number $N=rh$ was $$ \log C(rh)=\frac{1}{2\log r}\left(\log\frac{h}{\log h}\right)^2 +\left(\frac{1}{2}+\frac{1}{\log r}+\frac{\log\log r}{\log r}\right)\log h\\-\left(1+\frac{\log\log r}{\log r}\right)\log\log h+O(1)$$ De Bruijn went on to calculate the O(1) which was a periodic but not differentiable function.
$C(rh)$ obeys the functional equation $C(rh)=C(rh-r)+C(h)$.