Let $A$ be the number of ways in which the set $\{ 1, 2, . . . , n\}$ can be partitioned into non-empty subsets. Let $B$ be the number of ways in which the set $\{ 1, 2, . . . , n, n + 1 \}$ can be partitioned into non-empty subsets such that consecutive numbers belong to distinct subsets. Partitions that differ only in the order of the subsets are considered equal. Prove that $A = B$.