Suppose you have $M$ distinguishable objects distributed amongst $N$ distinguishable boxes. Can you calculate the expected maximum occupation number $E_\text{max}(N,M)$? (in other words, the expected number of objects in the box with the most objects in it?)
I'm having trouble even writing this down as a sum, a convoluted way to do it (I think) would be: $$E_\text{max}(M,N) = \frac{1}{{M+N-1}\choose{M}}\sum_{\lambda \vdash_\mathrm{ordered} n} m(\lambda) {N \choose |\lambda|} \frac{M!}{\prod_i \lambda_i!}$$ where we sum over all ordered partitions $\lambda_1 + \cdots \lambda_{|\lambda|} = n$. If this sum is hard, I'd also be interested in any asymptotics, I know in the limit that $N$ is very large and $M$ is fixed, I can write: $$E_\text{max}(M,N) = 1 + \frac{M(M-1)}{2(N-M+1)} + O\left(\frac{1}{N^2}\right)$$ Id be interested in seeing $E_\text{max}(N,N) \sim \log(N)$ though (I don't know if this is true)
