The Number of Distinct Part Sizes in a Random Integer Partition

A par t i t ion of n is a mul t i se t of positive integers whose sum is n. The summands , i.e., the e lements of the mult iset , are called parts. Let 9 , be the set of all par t i t ions of n, and let P ( n ) = [ 9 , [ . Put the un i fo rm probabi l i ty measure on ~ , ; mn({A}) = 1 / P ( n ) for all h ~ , . T h e n any real va lued func t ion X~ on ~ , can be regarded as a r a n d o m variable. If X n (n = 1 ,2 ,3 . . . . ) is a sequence of funct ions that arises natura l ly in combinator ics , then it is of ten reasonable to ask quest ions about the asymptotic d is t r ibut ion of values as n ~ ~. Erdds and Lehner [3] were apparen t ly the first to study r a n d o m integer par t i t ions in this way. Subsequen t work by a n u m b e r of authors provides considerable in format ion about the s t ructure of a " typical" part i t ion. (See, for example, Fr is tedt [4] and Szalay and Turfin [6].)