I have run across a question that seems like it should have a well known answer, but I can't find one, so I thought I would ask this hive mind:

Suppose we start with t piles of s rocks each. In a given turn, I will choose at random (with equal probabilities) one of the piles that still has at least one rock in it and remove one of the rocks from that pile. After I have removed k total rocks (with k less than st), what is the expected number of piles that are left? It would be even nicer to know what the probability is that a given number of piles remain or the expected size of the largest pile or things of that nature.

I have done some monte carlo simulations and I would be happy to share those results, but I am curious if anyone has any insights into this or has run across something similar in the literature.

Thanks!