Let $\bar p_k(n)$ be the number of partitions of $n$ with largest part at most $k$ (equivalently, into at most $k$ parts). Is there an elementary formula for the asymptotic behavior of $\bar p_k(n)$ as $n \rightarrow \infty$ that still involves $k$ in a nontrivial way?
I would also like to know of any references that discuss the numbers $\bar p_k(n)$. Browsing Stanley v.1 isn't much help, though it's possible that I am unaware of a section in the later parts or in v.2.