✦ LIBER ✦

On the multiplicity of parts in a random partition

✍ Scribed by Sylvie Corteel; Boris Pittel; Carla D. Savage; Herbert S. Wilf

Publisher: John Wiley and Sons
Year: 1999
Tongue: English
Weight: 200 KB
Volume: 14
Category: Article
ISSN: 1042-9832
DOI: 10.1002/(sici)1098-2418(199903)14:2<185::aid-rsa4>3.0.co;2-f

No coin nor oath required. For personal study only.

✦ Synopsis

Let be a partition of an integer n chosen uniformly at random among all Ž . such partitions. Let s be a part size chosen uniformly at random from the set of all part Ž . sizes that occur in . We prove that, for every fixed m G 1, the probability that s has Ž Ž .. multiplicity m in approaches 1r m mq 1 as n ª ϱ. Thus, for example, the limiting probability that a random part size in a random partition is unrepeated is 1r2. In addition, Ž . a for the average number of different part sizes, we refine an asymptotic estimate given by Ž . Wilf, b we derive an asymptotic estimate of the average number of parts of given Ž . multiplicity m, and c we show that the expected multiplicity of a randomly chosen part size

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