How many partitions of $n$ distinct objects are there, given that all parts are equinumerous? (Let us not consider the empty partition and the identity partition.)
The cases when $n$ is the unit, or when $n$ is prime, are trivial. In these cases we have $0$ partition and $1$ partition respectively.
So treat this question as asking about the number of such partitions for composite $n.$
In particular, the case when $n = p^2$ with $p$ prime is very interesting. For example, when $n = 4,$ we have $3$ such partitions. What of when $n$ is $9,$ or $25,$ or $49,$ and so on?
Thank you.
Best Answer
Number of ways to partition an $n$-set into subsets of equal size is tabulated at http://oeis.org/A038041 where there are formulas, programs, and links to the literature.