# Count all possible unions in a collection of sets

Say we have a collections of sets $\mathcal X = \{X_1, \dots, X_n\}$ (not necessarily disjoint), and we want to count the number of possible unions of sets in $\mathcal X$, i.e. the size of $\{\bigcup_{Y \in \mathcal Y} Y \,|\, \mathcal Y \subseteq \mathcal X \}$.

Is there a name for this problem? What are the fastest algorithms to solve this problem?