Question 28
Let $S=\{1,2, \ldots, n\}$, and let $T$ be the set consisting of all nonempty subsets of $S$. The function $f: T \rightarrow S$ is "garish" if there do not exist sets $A, B \in T$ such that $A$ is a proper subset of $B$ and $f(A)=f(B)$. Determine, with proof, how many garish functions exist.