On the Distribution of the Longest Run in Number Partitions

We consider the distribution of the longest run of equal elements in number partitions (equivalently, the distribution of the largest gap between subsequent elements); in a recent paper, Mutafchiev proved that the distribution of this random variable (appropriately rescaled) converges weakly. The corresponding distribution function is closely related to the generating function for number partitions. In this paper, this problem is considered in more detail—we study the behavior at the tails (especially the case that the longest run is comparatively small) and extend the asymptotics for the distribution function to the entire interval of possible values. Additionally, we prove a local limit theorem within a suitable region, i.e. when the longest run attains its typical order n1/2, and we observe another phase transition that occurs when the largest gap is of order n1/4: there, the conditional probability that the longest run has length d, given that it is ≤ d, jumps from 1 to 0. Asymptotics for the mean and variance follow immediately from our considerations.