On the multiplicity of parts in a random partition

Let lambda be a partition of an integer n chosen uniformly at random among all such partitions. Let s(lambda) be a part size chosen uniformly at rando m from the set of all part sizes that occur in lambda. We prove that, for e very fixed m greater than or equal to 1, the probability that s(lambda) has multiplicity m in A approaches 1/(m(m + 1)) as n ---> infinity. Thus, for example, the limiting probability that a random part size in a random parti tion is unrepeated is 1/2. In addition, (a) for the average number of diffe rent part sizes, we refine an asymptotic estimate given by Wilf, (b) we der ive an asymptotic estimate of the average number of parts of given multipli city m, and (c) we show that the expected multiplicity of a randomly chosen part size of a random partition of n is asymptotic to (log n)/2. The proof s of the main result and of (c) use a conditioning device of Fristedt. (C) 1999 John Wiley & Sons, Inc.