Let \mathcal{P} be the power set of \{1,2,3,4\} (meaning the elements of \mathcal{P} are the subsets of \{1,2,3,4\} ). How many subsets S of \mathcal{P} are there such that no two distinct integers a, b \in\{1,2,3,4\} appear together in exactly one element of S?