Is there a general way to count the number of sigma-algebras on a finite set? I was asked a high-school question today which was merely asking the number of sigma-algebras on a set.

Let $X$ be a set
Let $S\triangleq \{\Sigma\subset P(X): \Sigma \text{ is a sigma-algebra on }X\}$.
If $X$ is finite, what would be the cardinality of $S$?

The question i was asked was the case $|X|=3$. In this case $|S|=5$.
What would be the cardinality of $S$ in general? Is it possible to find this value via elementary function?

A sigma algebra on a finite set is uniquely determined by the minimal non-empty elements under inclusion. (Why?) 
The minimal elements are pair-wise disjoint. (Why?) 
The minimal elements must have the entire set as a union. (Why?)
So the minimal elements form a partition of the set.
