Number of combinations for creating subsets including all items

Question

Say I have n items. How can I find the amount of combinations possible of subset sets covering all items, when order does not matter. This is maybe badly worded, so allow me to give an example:

n=3, so we have items A,B, and C. All possible subset sets covering all items would be - [A B C] - [A BC] - [AC B] - [AB C] - [ABC] That makes for 5 possibilities.

Say the function to calculate this is f(n) and the total amount of subsets in all sets is g(n) (that would mean g(3) as in the example is 10). I managed to figure out f(n)=f(n-1)+g(n-1). However, this is where I am kind of stuck.

Answer 1

These are counted by Bell Numbers. Unfortunately there is no explicit formula for $f(n)$, but there are plenty of recursions. For example:

$$f(n+1)=\sum_{k=0}^n\binom{n}{k}f(k).$$