We analyze the asymptotic behavior of the average maximal number of balls in a bin obtained by throwing uniformly at random $r$ balls without replacement into $n$ bins, $T$ times. Writing the expected maximum as $\frac{ r}{ n}T+ C_{ n,r}\sqrt{ T} + o(\sqrt{ T})$, a recent preprint of Behrouzi-Far and Zeilberger asks for an explicit expression for $C_{ n,r}$ in terms of $n,r$ and $\pi$. In this short note, we find an expression for $C_{ n,r}$ in terms of $n, r$ and the expected maximum of $n$ independent standard Gaussians. This provides asymptotics for large $n$ as well as closed forms for small $n$---e.g. $C_{ 4,2} = \frac{ 3}{ 2 \pi^{ 3/2}} \arccos(-1/3)$---and shows that computing a closed form for $C_{ n,r}$ is precisely as hard as the difficult question of finding the expected maximum of $n$ independent standard Gaussians.
Categories: math.CO math.PR