Originally Posted by
Plato In point of fact, the Bell numbers are given by $\displaystyle B(n) = \sum\limits_{k = 0}^n {S_2 (n,k)} $ .
Now $\displaystyle {S_2 (n,k)}$ are the Stirling numbers of the second kind.
That is the number of ways to partition a set of n elements into k nonempty subsets.
To calculate: $\displaystyle {S_2 (n,k)}= \frac{1}{{k!}}\left[ {\sum\limits_{k = 0}^n {\left( { - 1} \right)^j \binom{k}{j}\left( {k - j} \right)^n } } \right]$.
So the Bell numbers count the total number of ways to partition a set.