Bell Numbers

The Bell numbers count the possible partitions of a set, that is, the number of ways to split its elements into nonempty groups regardless of order. They have been studied by mathematicians since the nineteenth century, with roots going back to medieval Japan, and in an example of Stigler's law of eponymy they are named after Eric Temple Bell, who wrote about them in the 1930s. Written B(n) for a set of n elements, with B(0) and B(1) both equal to one, the sequence begins 1, 1, 2, 5, 15, 52, 203, 877, 4140 and so on. Beyond counting partitions and equivalence relations on a set, and the number of rhyme schemes for an n line poem, the Bell numbers also have a probabilistic interpretation: B(n) is the n-th moment of a Poisson distribution with mean one.