APMO 2014 - Asian Pacific Mathematics Olympiad Question 2

Let $S=\{1,2,\ldots,2014\}$. Choose one member of each nonempty subset $T\subseteq S$ as its representative. Count the assignments such that whenever $D$ is a disjoint union of three nonempty subsets $A,B,C$, the representative of $D$ is also the representative of at least one of $A,B,C$.

Proposed by Warut Suksompong, Thailand.