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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):
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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.)
The Bell numbers satisfy this recursion formula:
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They also satisfy "Dobinski's formula":
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And they satisfy "Touchard's congruence": If p is any prime
number then
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Each Bell number is a sum of "Stirling numbers of the second
kind"
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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 n cumulants; this way of enumerating partitions is not as coarse as that given by the Stirling numbers. See Bell polynomials for more on the connection to cumulants.
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