# Possible count of solutions for clustering $n$ objects

Assuming $n$ objects are given. I like to get the number of possible solutions for a clustering algorithm. Obviously the cluster count must be greater or equal to $1$ and smaller or equal to $n$. For these two cases it is also easy to calculate how many solutions exists: For both exactly one.

If it is assumed there are $2$ clusters it is also not too hard: There are (probably) $2^n-2$ possible solutions.

I like to add up the amount of all solutions, but I fail to generalize the count of solutions for a specific amount of clusters (if this is given, then it should be easy).

Maybe someone knows how to do this?

Thank you very much

• Search for Stirling numbers of the second kind. – Phicar Jun 25 '17 at 21:57
• Cool, thanks:)! – Kevin Meier Jun 25 '17 at 22:09