Bell number

The Bell number Bₙ counts the partitions of an n-element set into nonempty, disjoint subsets. It begins 1, 1, 2, 5, 15, 52 for n = 0, 1, 2, 3, 4, 5.

Bell triangle: A triangular array whose row sums are the Bell numbers. Adjacent entries generate each new row, giving a direct recurrence for the sequence.
Set partition: A division of a set into nonempty, pairwise disjoint subsets whose union is the whole set. Bell numbers count these objects across every possible number of blocks.
Random partition: A probability distribution over the partitions of a set. The Bell number gives the size of the sample space when all partitions are possible.
Eric Temple Bell: A Scottish-born mathematician known for work in number theory and the history of mathematics. Bell numbers were named in his honor, though the underlying sequence predates him.
Dobinski's formula: The identity Bₙ = e⁻¹∑ₖ₌₀∞ kⁿ/k! for the nth Bell number. It expresses a finite set-partition count as an infinite series.
Equivalence relation: A relation that is reflexive, symmetric, and transitive. Each set partition corresponds uniquely to an equivalence relation on the same set.
Chinese restaurant process: A sequential random process that assigns arriving items to existing or new groups. Its possible groupings of n arrivals are set partitions counted in total by Bₙ.
Combinatorics: The branch of mathematics concerned with counting, arrangement, and discrete structures. Bell numbers became a central counting sequence within enumerative combinatorics.
Exponential generating function: A power series whose coefficient of xⁿ/n! encodes the nth term of a sequence. The Bell numbers have generating function exp(eˣ − 1), which yields useful identities.
Partition lattice: The partially ordered set of partitions of a finite set, ordered by refinement. Its elements are precisely the objects counted by a Bell number.

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