2014 APMO 第 2 题

Let $S=\{1,2, \ldots, 2014\}$. For each non-empty subset $T \subseteq S$, one of its members is chosen as its representative. Find the number of ways to assign representatives to all non-empty subsets of $S$ so that if a subset $D \subseteq S$ is a disjoint union of non-empty subsets $A, B, C \subseteq S$, then the representative of $D$ is also the representative of at least one of $A, B, C$. (Warut Suksompong, Thailand)

设$S=\{1,2,\ldots, 2014\}$。对于每个非空子集 $T \subseteq S$，选择其成员之一作为其代表。求将代表分配给 $S$ 的所有非空子集的方法的数量，使得如果子集 $D \subseteq S$ 是非空子集 $A, B, C \subseteq S$ 的不相交并集，则 $D$ 的代表也是 $A, B, C$ 中至少一个的代表。 (Warut Suksompong，泰国)