Sampling parts of random integer partitions: a probabilistic and asymptotic analysis

Let λ 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 λ at random. They obtained limiting distributions of the multiplicity μ_n = μ_n(λ) of the randomly-chosen part as n → ∞. The asymptotic behavior of the part size σ_n = σ_n(λ), 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 (μ_n; σ_n), as n → ∞, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for (μ_n; σ_n). Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of μ_n and σ_n.