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