Limit distributions of smallest gap and largest repeated part in integer partitions

Abstract

We study two parameters in random integer partitions, namely the first gap and the last repeated part, that have been introduced by Grabner and Knopfmacher in a recent paper (Ramanujan J. 12(3):439–454, 2006). More generally, the first part that occurs at most r times and the last part that occurs at least r times are considered. For both parameters, we determine the limit distribution, which turn out to be the Rayleigh and Gumbel distributions, respectively. This also generalises the well-known result by Erdős and Lehner on the distribution of the largest part in a random integer partition. Furthermore, extensions to general Λ-partitions and results on related parameters, such as the length of the first gap, are provided.