There are $n$ stones distributed in $n$ buckets (initially one stone in each bucket). At each step the content of each bucket is put in a random bucket, chosen independently out of a set of $n$ new empty buckets. Doing this the groups of stones in the differents buckets may merge together.
Now, after $i$ steps we consider the non-empty groups of stones, and I was wondering if there is any information about the (two dimensional) distribution (giving a probability of having at least $\geq b$ buckets with $\geq k$ stones, say), in particular if the problem was studied for big $n$ and big $i$ relative to $n$. Was this model studied before?
The problem arises from simulations of discretized chaotic dynamical systems, where this model (could) provide a comparison model, for long time behaviour and fine discretization.