On the multiplicity of parts in a random partition

Let be a partition of an integer n chosen uniformly at random among all such partitions. Let s( ) be a part size chosen uniformly at random from the set of all part sizes that occur in . We prove that, for every xed m 1, the probability that s( ) has multiplicity m in approaches 1=(m(m + 1)) as n ! 1. Thus, for example, the limiting probability that a random part size in a random partition is unrepeated is 1/2. In addition, (a) for the average number of di erent part sizes, we re ne an asymptotic estimate given by Wilf, (b) we derive an asymptotic estimate of the average number of parts of given multiplicity m, and (c) we show that the expected multiplicity of a randomly chosen part size of a random partition of n is asymptotic to (logn)=2. The proofs of the main result and of (c) use a conditioning device of Fristedt. AMS Subject Classi cations: 05A17, 11P81, 11P82 Supported in part by NSF grant DMS9302505 Supported by the Focus on Discrete Probability Program at Dimacs Center of Rutgers University, and by NSA grant MDA-904-96-1-0053 Supported in part by NSF grant DMS96227722