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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

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    You want $k$ **equal** parts? What about $[k\mid n]$ (using the [Iverson bracket](http://en.wikipedia.org/wiki/Iverson_bracket)).2012-11-19
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    No I do not form not equal parts. I am looking for the analogous formula for number of compositions in k parts which is $\binom{n-1}{k-1}$.2012-11-19
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    You have a cryptic way of expressing yourself. Saying you "do _not_ form _not equal_ parts" means you **do** form **equal parts only**. I dont't think that is what you mean, but it is what you say. And the question says equal parts too.2012-11-19
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    I'm sorry. I do not want equal parts. That was a mistake in typing.2012-11-19

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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 $k$.

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    Thank you. Is a formula for a general $k$ known on the lines of the analogous formula for compositions $\binom{n-1}{k-1}$?2012-11-22
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    Ask me again, after you have looked at Andrews and Eriksson.2012-11-22
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    I have looked at the reference. The basic idea is using the corresponding generating function. However I am still in doubt whether a general formula, howsoever complicated but in closed form, is known or not?2012-11-22
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    Well, it seems unlikely. They have to use different methods for different $k$, and the formulas look very different, and they don't say anything about there being a general formula. I deduce from that that there isn't a general formula, but I recognize that's not a proof.2012-11-22