Proof for identity for bell numbers

Question

How can I proof this identity for bell numbers?

$$B_n = \sum_{k=0}^n S(n,k)$$

Is it possible without using the recurrence relation?

Answer 1

It is clearly true if you define:

• $B_n$ to be the number of ways to partition a set of cardinality $n$ into non-empty subsets; and
• $S(n,k)$ to be the number of ways to partition a set of cardinality $n$ into exactly $k$ non-empty subsets

Any partition counted in $B_n$ is counted once in exactly one of the $S(n,k)$. Similarly any partition counted in any of the $S(n,k)$ is counted exactly once in $B_n$.

$k$ cannot be less than $0$ or more than $n$. So $$B_n = \sum_{k=0}^{n} S(n,k). \qquad \qquad \square$$