Limit theorems for the number of parts in a random weighted partition
type: Item
instance of: scholarly article
title: Limit theorems for the number of parts in a random weighted partition
zbMATH Open document ID: 1238.05019
published in: The Electronic Journal of Combinatorics
publication date: 2011-11-22T00:00:00Z
full work available at URL: v18i1p206.html
review text: 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.
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 5976484
author: Ljuben Mutafchiev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5976484