TITLE: Asymptotics for partitions of $n$ with largest part at most $k$ (or into at most $k$ parts)
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: Very useful paper, though this only gives a good approximation to the number of partitions of n
into exactly k parts each no larger than N
due to Ratsaby (App. Analysis and Discrete M. 2008): 
http://www.doiserbia.nb.rs/img/doi/1452-8630/2008/1452-86300802222R.pdf