Partitions of an integer into k parts.

I am interested in knowing whether an exact formula (analogous to the Hardy-Ramanujan-Rademacher formula for $p(n)$) for the number of partitions of a positive integer into k parts is known. I tried searching on the internet but could not find such a formula. Can someone confirm whether such a formula is known and provide a link to its proof if possible.

Thanks

In Chapter 6 of Andrews and Eriksson, Integer Partitions, formulas are given for $k=1,2,3,4,5$. Full proofs are given for $k=1,2,3,4$, while $k=5$ is left as an exercise.

[EDIT: By the way, this is a very nice little book. "The aim of this introductory textbook is to provide an accessible and wide-ranging introduction to partitions, without requiring anything more of the reader than some familiarity with polynomials and infnite series."]

Actually, the function studied in that text is a bit different from what you want, but it's easy to get back and forth between the two functions. The number of partitions into (exactly) $k$ parts equals the number of partitions with greatest part $k$. The text studies the number of partitions with greatest part at most $k$. But partitions of $n$ with greatest part $k$ correspond to partitions of $n-k$ with greatest part at most $k$.