On the distribution of the largest part of unrestricted partitions of small integers

Abstract

Several theoretical estimates of the distribution of the parts of integer partitions have been published. Generally these are asymptotically correct for large integers, but practical applications require that the distribution be known for small integers (n £ 1000).  The largest part (or the number of parts) of an unrestricted partition of the integer n has the extreme value distribution, in agreement with the theoretical estimates. Expressions approximating the mode and variance of the distribution are given for n £ 1000 that represent significant improvements over the asymptotically correct theoretical expressions.