Sampling Parts of Random Integer Partitions: A Probabilistic and Asymptotic Analysis

TL;DR

This paper investigates the joint asymptotic distribution of the size and multiplicity of a randomly chosen part in a uniform random integer partition, revealing how different sampling methods influence these distributions.

Contribution

It provides the first comprehensive analysis of the joint limiting distribution of size and multiplicity under various sampling procedures for random partitions.

Findings

Different sampling procedures yield distinct joint limiting distributions.

Results confirm and extend previous marginal distribution findings.

Joint distributions depend on the sampling method used.

Abstract

Let $\lambda$ be a partition of the positive integer $n$, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of $\lambda$ at random. They obtained limiting distributions of the multiplicity $\mu_n=\mu_n(\lambda)$ of the randomly-chosen part as $n\to\infty$. The asymptotic behavior of the part size $\sigma_n=\sigma_n(\lambda)$, under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partition. We describe it obtaining the joint limiting distributions of $(\mu_n,\sigma_n)$, as $n\to\infty$, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for $(\mu_n,\sigma_n)$. Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of $\mu_n$ and $\sigma_n$.