On the Largest Part Size and Its Multiplicity of a Random Integer Partition

TL;DR

This paper investigates the asymptotic behavior of the largest part size and its multiplicity in a random integer partition, revealing their similar distributional limits but differing growth rates in expectation.

Contribution

It provides a detailed asymptotic analysis of the expectation difference between the largest part size times its multiplicity and the largest part size alone in random partitions.

Findings

Distributional similarity of L_n and L_n M_n for large n

Expectation of L_n M_n - L_n grows as (1/2) log n

Asymptotic expansion for the expectation difference

Abstract

Let $\lambda$ be a partition of the positive integer $n$ chosen umiformly at random among all such partitions. Let $L_n=L_n(\lambda)$ and $M_n=M_n(\lambda)$ be the largest part size and its multiplicity, respectively. For large $n$, we focus on a comparison between the partition statistics $L_n$ and $L_n M_n$. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of $L_n M_n -L_n$ grows as fast as $\frac{1}{2}\log{n}$ We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.