On the distribution of the longest run in number partitions

Abstract

Abstract. 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 n 1/2 , and we observe another phase transition that occurs when the largest gap is of order n 1/4 : 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.

A General Asymptotic Scheme for the Analysis of Partition Statistics

“…The technique to obtain a very precise asymptotic expansion is similar to the previous example, but the details are somewhat more intricate. The final result is stated in the following proposition; a weak version (first-order asymptotics only) was given in [25].…”

“…The longest run (maximum multiplicity, i.e., the greatest number of times a part gets repeated in a partition) was shown to follow a rather unusual limit law in [17]; see also [25], where the mean of the longest run was found to be asymptotically equal to…”

“…It is actually possible to determine the asymptotic behaviour of all moments and thus to determine the limiting distribution (different approaches were used in [17] and [25]). (1 − e −πjx/ √ 6 ).…”

Integer partitions, probabilities and quantum modular forms

“…Notable in those are the works dealing with runs and gaps in parts making up a partition [17,18,20,22,48,51,57,78,79]. This paper demonstrates how quantum modular forms and their asymptotic expansions, which involve values of modular L-functions, arise in the resolution of natural integer partition probability problems.…”