Gian-Carlo Rota, The Number of Partitions of a Set, The American Mathematical Monthly, Vol. 71, No. 5 (May, 1964), pp. 498-504 ...

Let P(n), Q(n) denote, respectively, the set of all unrestricted partitions of n and the set of all partitions of n into distinct parts. For 1 ≤ j ≤ n, we derive formulas that permit the computation ...