Bell number

In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s. The Bell numbers are denoted , where is an integer greater than or equal to zero. Starting with , the first few Bell numbers are 1, 1, 2, 5, 15, 52, 203, 877, 4140, ... . The Bell number counts the number of different ways to partition a set that has exactly elements, or equivalently, the number of equivalence relations on it. also counts the number of different rhyme schemes for -line poems. As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, is the -th moment of a Poisson distribution with mean 1. Partition of a set In general, is the number of partitions of a set of size . A partition of a set is defined as a family of nonempty, pairwise disjoint subsets of whose union is . For example, because the 3-element set can be partitioned in 5 distinct ways: As suggested by the set notation above, the ordering of subsets within the family is not considered; ordered partitions are counted by a different sequence of numbers, the ordered Bell numbers. is 1 because there is exactly one partition of the empty set. This partition is itself the empty set; it can be interpreted as a family of subsets of the empty set, consisting of zero subsets. It is vacuously true that all of the subsets in this family are non-empty subsets of the empty set and that they are pairwise disjoint subsets of the empty set, because there are no subsets to have these unlikely properties. The partitions of a set correspond one-to-one with its equivalence relations. These are binary relations that are reflexive, symmetric, and transitive. The equivalence relation corresponding to a partition defines two elements as being equivalent when they belong to the same partition subset as each other.