Recurrence Relation for Bell Numbers

Theorem
Let $B_n$ be the Bell number for $n \in \Z_{\ge 0}$.

Then:
 * $B_{n + 1} = \ds \sum_{k \mathop = 0}^n \dbinom n k B_k$

where $\dbinom n k$ are binomial coefficients.

Proof
By definition of Bell numbers:


 * $B_{n + 1}$ is the number of partitions of a (finite) set whose cardinality is $n + 1$.

Let $k \in \set {k \in \Z: 0 \le k \le n}$.

Let us form a partition of a (finite) set $S$ with cardinality $n + 1$ such that one component has $n + 1 - k > 0$ elements.

We can do this by first choosing $1$ element from $S$. We put this element into that single component.

Then choose $k$ more elements from $S$, and let the remaining $n - k$ elements be put into the same component as the first element.

From Cardinality of Set of Subsets and the definition of binomial coefficient, there are $\dbinom n k$ ways to do this.

For the chosen $k$ elements, there are $B_k$ ways to partition them.

Thus there are $\dbinom n k B_k$ possible partitions for $S$:
 * $\dbinom n k$ of selecting $n - k$ elements to form one component with the one singled-out element
 * for each of these, $B_k$ ways to partition the remaining $k$ elements.

Summing the number of ways to do this over all possible $k$:


 * $\ds B_{n + 1} = \sum_{k \mathop = 0}^n \dbinom n k B_k$