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?



Distribution of colors in the number of integer partitions of n
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 
$n^{3/4}$. 


