{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:04Z","timestamp":1753893844085,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.37236\/693","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:40:53Z","timestamp":1578714053000},"source":"Crossref","is-referenced-by-count":3,"title":["Limit Theorems for the Number of Parts in a Random Weighted Partition"],"prefix":"10.37236","volume":"18","author":[{"given":"Ljuben","family":"Mutafchiev","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2011,10,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p206\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p206\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:00:41Z","timestamp":1579302041000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p206"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,24]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/693","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,10,24]]},"article-number":"P206"}}