Let $m$ and $n$ be two positive integers and denote by $P(n,m)$ the number of partitions of $n$ into $m$ non-negative integers.
Is there an asymptotic formula for $P(n,m)$ ?? Any reference is welcome. Thanks
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1$\begingroup$ What do you mean when you say asymptotic formula? (It could be fixed $m$ and $n$ tends to infinity, or they tend to infinity by a different rate... There are many possibilities!) $\endgroup$– Daniel SoltészCommented Jan 2, 2015 at 22:03
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$\begingroup$ @Soltész I would like arbitrary $m$ and $n$; is this a very hard question? $\endgroup$– GiulioCommented Jan 5, 2015 at 9:55
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1$\begingroup$ Well, that's not called an asymptotic formula. I am not an expert in this part of mathematics, but based on the results known to me, I think that this is a very hard question. $\endgroup$– Daniel SoltészCommented Jan 5, 2015 at 10:21
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2 Answers
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For $m\geq n^{1/6}$ there is an asymptotic formula due to Szekeres. See http://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r6/pdf for references and another proof.
answered Jan 2, 2015 at 22:17
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$\begingroup$ @Stanley is there a formula without asking any relation between $m$ and $n$? $\endgroup$– GiulioCommented Jan 5, 2015 at 9:54
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In the reference cited below, the following asymptotic formula is provided when $m < n\le 2m$:
$$p(n,m) \approx \frac{1}{4\sqrt{3}(n-m)}e^{\pi\sqrt{\frac{2(n-m)}{3}}}$$.
It's also shown in the same reference that
$$p(n,m) \le \frac{5.44}{(n-m)}e^{\pi\sqrt{\frac{2(n-m)}{3}}}, 1\le m \le n-1.$$
A. Y. Oruç. "On number of partitions of an integer into a fixed number of positive integers." Journal of Number Theory 159 (2016): 355-369.
