How many ways are there to partition $n$ unique elements into $2$ sets? What about for $k$ sets?
I am specifically interested in how to calculate this for varying values of $n$. Moreover, what if there are restrictions, such as the sum of elements in one set must be equal to the sum of the elements in the other?
These are the Stirling coefficients of the second kind. If the sum of elements in sets must be equal the problem is a lot harder.