Limit theorems for the number of parts in a random weighted partition
Summary: Let \(c_{m,n}\) be the number of weighted partitions of the positive integer \(n\) with exactly \(m\) parts, \(1\leq m\leq n\). For a given sequence \(b_k\), \(k\geq 1\), of part type counts (weights), the bivariate generating function of the numbers \(c_{m,n}\) is given by the infinite product \[ \prod^\infty_{k=1} (1- uz^k)^{-b_k}. \] Let \(D(s)=\sum^\infty_{k=1} b_k k^{-s}\), \(s= \sigma+ iy\), be the Dirichlet generating series of the weights \(b_k\). In this present paper we consider the random variable \(\xi_n\) whose distribution is given by \[ P(\xi_n= m)= c_{m,n}\Biggl(\sum^n_{m=1} c_{m,n}\Biggr), \] \(1\leq m\leq n\). We find an appropriate normalization for \(\xi_n\) and show that its limiting distribution, as \(n\to\infty\), depends on properties of the series \(D(s)\). In particular, we identify five different limiting distributions depending on different locations of the complex half-plane in which \(D(s)\) converges.
scientific article; zbMATH DE number 5976484