THE NUMBER OF DISTINCT PART SIZES IN A RANDOM INTEGER PARTITION

We prove a central limit theorem for the number of different part sizes in a random integer partition. If lambda is one of the P(n) partitions of the integer n, let D-n(lambda) be the number of distinct part sizes that lambda has. (Each part size counts once, even though there may be many parts of a given size.) For any fixed x, #{lambda: D-n(lambda) less than or equal to A(n) + B-n}/P(n) --> 1/root 2 pi integral(-infinity) (x) e(-t2/2)dt as n --> infinity, where A(n) = root 6/pi)n(1/2) and B-n = (root 6/2 pi - root 54/pi(3))(1/2)n(1/4). (C) 1995 Academic Press, Inc.