On the Average Maximal Number of Balls in a Bin Resulting from Throwing r Balls into n Bins T times

Abstract

We use the holonomic ansatz to estimate the asymptotic behavior, in $T$, of the average maximal number of balls in a bin that is obtained when one throws uniformly at random (without replacement) $r$ balls into $n$ bins, $T$ times. Our approach works, in principle, for any fixed $n$ and $r$. We were able to do the cases $(n,r)$ = $(2,1),(3,1),(4,1), (4,2)$, but things get too complicated for larger values of $n$ and $r$. We are pledging a \$150 donation to the OEIS for an explicit expression, (in terms of $n$, $r$, and $\pi$) for the constant $C_{n,r}$ such that that average equals $\frac{n}{r}\,T+C_{n,r} \sqrt{T}+O(1/\sqrt{T})$. In this version we announce that the problem has been solved (to the extent possible) by Marcus Michelen.