Limit Theorems for the Number of Parts in a Random Weighted Partition

Abstract

Let $c_{m,n}$ be the number of weighted partitions of the positive integer $n$ with exactly $m$ parts, $1\le m\le n$. For a given sequence $b_k, k\ge 1,$ of part type counts (weights), the bivariate generating function of the numbers $c_{m,n}$ is given by the infinite product $\prod_{k=1}^\infty(1-uz^k)^{-b_k}$. Let $D(s)=\sum_{k=1}^\infty b_k k^{-s}, s=\sigma+iy,$ 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 $P(\xi_n=m)=c_{m,n}/(\sum_{m=1}^nc_{m,n}), 1\le m\le n$. 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.