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

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 $\sigma_n$.