In light of this question, Counting seating arrangements at a circular table, I can't help but wonder how one would count the number of seating arrangements we could make if, instead of 1 table of size $n$, we had $d$ equally sized tables where $d|n$. That is, if we have $b\lt n−1$ identitcal boys and we divide the boys into k teams of size ≥1, how many ways are there to seat the boys at $d$ tables of size $p=n/d$ such that the teams either sit at different tables or, if they are at the same table there is at least 1 empty seat separating each team from the other teams at the table? Consider the seats to be unique (e.g. if we have only 1 boy and 1 table, there are n possible seating arrangements for him).
We can rephrase this question in the way that it was phrased in the answer to the question mentioned above as follows: given $d$ rings of $p$ zeros, how many ways are there to replace $b$ of the zeros with 1's such that the 1's appear in $k$ "streaks" and each "streak" of 1's is separated from the others by at least one $0$?
In both of the above cases, we are limiting the size of our teams/"streaks" to $p$. That is, no team can occupy more than 1 table.
Thanks for you help!