How many of the partitions of a natural number $n$ are comprised only of its divisors? That is, if $$p(n)=\sum\_{\sum\_{1}^n kj\_k=n:j\_k\geq 0} 1\_{\[j\_1,j\_2,...\]},$$
is the ordinary partition function (i.e. the total number of partitions of $n$), then I want to know something about the counting function $$s(n)=\sum\_{\sum\_{d|n}dj\_d=n:j\_d\geq 0}1\_{[j\_1,...,j\_d,...]}.$$


I would be happy to hear of anything that is known about this function, but I am particularly interested in (a) its generating functions, and (b) a bijection between this restricted partition and another (hopefully more intuitive to count) restricted partition. Any insights would be welcome. 


I should add that google finds a number of papers that study "partitions of $n$ into divisors of $m$", e.g. Gupta, 1970s, but those methods reduce to rather vacuous statements when evaluated at $m=n$.


Thanks!



Partitions comprised only of divisors
Bounds for this partition function were given by the editors of The American Mathematical Monthly, Paul Erdos, and Andrew Odlyzko in the March 1992 issue, p. 277, as a solution to Advance Problem number 6640. The bounds they prove are:


$$
({\tau(n)}/{2}-1)(\log n + O({\log n}/{\log \log n}))
\leq 
\log s(n)
\leq
({\tau (n)}/{2})\log n + O(\log \log n),
$$
where $\tau(n)$ is the number of divisors of $n$.


