The Bell numbers, named in honor of Eric Temple Bell, are a sequence of integers arising in combinatorics that begins thus (sequence A000110 in OEIS):
In general, Bn is the number of partitions of a set of size n. (B0 is 1 because there is exactly one partition of the empty set. A partition of a set S is by definition a set of nonempty pairwise disjoint sets whose union is S. Every member of the empty set is a nonempty set (that is vacuously true), and their union is the empty set. Therefore, the empty set is the only partition of itself.)
And they satisfy "Touchard's congruence": If p is any prime number then
Each Bell number is a sum of "Stirling numbers of the second kind"
The Stirling number S(n, k) is the number of ways to partition a set of cardinality n into exactly k nonempty subsets.
The nth Bell number is also the sum of the coefficients in the polynomial that expresses the nth moment of any probability distribution as a function of the first ncumulants; this way of enumerating partitions is not as coarse as that given by the Stirling numbers.
In combinatorial mathematics, the nth Bellnumber, named in honor of Eric Temple Bell, is the number of partitions of a set with n members, or equivalently, the number of equivalence relations on it.
The nth Bellnumber is also the sum of the coefficients in the polynomial that expresses the nth moment of any probability distribution as a function of the first n cumulants; this way of enumerating partitions is not as coarse as that given by the Stirling numbers.
The number in the nth row and kth column is the number of partitions of {1,..., n} such that n is not together in one class with any of the elements k, k + 1,..., n − 1.
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