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Given an integer $n$ the number of partitions of $n$ into two colors can be represented as $$p_2(n)=\sum_{k=0}^n p(k)p(n-k)$$ where $p(k)$ counts the number of ordinary partitions of $k.$ What is the distribution of $$P(k)=\frac{p(k)p(n-k)}{p_2(n)}$$ as $n\to \infty.$

I feel as though this question has probably been addressed in the past but I am unaware of where in the literature. Does anyone know where I can find this?

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$P(k)$ is maximal for $k$ around $n/2$, of course, and is concentrated therein. Specify, what exactly asymptotical question are you interested in. – Fedor Petrov May 3 '13 at 22:53
I'm more interested in a sharp concentration inequality or for that matter a place where I can reference it. – Daniel Parry May 3 '13 at 23:05
up vote 1 down vote accepted

The Hardy-Ramanujan asymptotic formula gives $p(n) \sim \exp(C\sqrt{n})/(4n\sqrt{3})$ with $C= \pi \sqrt{2/3}$. From this one sees that if $\ell$ is not too big then $$ p(n+\ell)p(n-\ell) \sim p(n)^2 \exp(C (\sqrt{n+\ell}+\sqrt{n-\ell} -2\sqrt{n}) ) \sim p(n)^2 \exp\Big( -C \frac{\ell^2}{4n^{3/2}} \Big). $$

Thus your function is concentrated around $k=n/2$ in an interval of size about $n^{3/4}$.

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