On the distribution of multiplicities in integer partitions

Paper presented at Strathmore International Math Research Conference on July 23 - 27, 2012We study the distribution of the number of parts of given multiplicity (or equivalently ascents of given size) in integer partitions. In this paper we give methods to compute asymptotic formulas for the expected value and variance of the number of parts of multiplicity d (d is a positive integer) in a random partition of a large integer n and also prove that the limiting distribution is asymptotically normal for fixed d. However, if we let d increase with n, we get a phase transition for d around n1=4. Our methods can also be applied to so called -partitions where the parts are members of a sequence of integers