AT-OEMO-FED1 2010 Problem 3

Given is the set $M_n=\{0, 1, 2, \ldots, n\}$ of nonnegative integers less than or equal to $n$ . A subset $S$ of $M_n$ is called outstanding if it is non-empty and for every natural number $k\in S$ , there exists a $k$ -element subset $T_k$ of $S$ . Determine the number $a(n)$ of outstanding subsets of $M_n$ .

(41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 3)