Sampling part sizes of random integer partitions

Abstract

Let $\lambda$ be a partition of the positive integer $n$, selected uniformly at random among all such partitions. Corteel et al. (Random Stuct Algorithm 14:185–197, 1999) proposed three different procedures of sampling parts of $\lambda$ at random. They obtained limiting distributions of the multiplicity of the randomly chosen part as $n\rightarrow \infty$. This motivated us to study the asymptotic behavior of the part size under the same sampling conditions. A limit theorem whenever the part is selected uniformly at random among all parts of $\lambda$ (i.e., without any size bias) was proved earlier by Fristedt (Trans Am Math Soc 337:703–735, 1993). We consider the remaining two (biased) procedures and show that in each of them the randomly chosen part size, appropriately normalized, converges in distribution to a continuous random variable. It turns out that different sampling procedures lead to different limiting distributions.