Suppose that we are given $n$ groups of $m$ people. We want to rearrange these $nm$ people into the same format of $n$ groups of $m$ with that the catch that any two people who were originally in a group together cannot be rearranged in the same group. How many ways can we rearrange the groups? It appears that this question is related to derangements but quite a bit more generalized.
A possibly related problem is the maximum number of times these rearrangements can happen before two previously encountered partners must meet again.
I am not sure if there is an elementary solution to these problems. It would appear that the first problem at least should be quite common in combinatorics and so I wonder if it is the special case of a famous problem.