Counting & Probability · Superchallenge problem

In how many ways can the set $\{1,2,3,4,5,6\}$ be partitioned into nonempty blocks so that no block contains two consecutive integers? (The blocks are unlabeled; for example $\{1,3\},\{2,5\},\{4,6\}$ is one such partition. The Bell numbers $B_1,\dots,B_6$ are $1,2,5,15,52,203$.)