{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,1,26]],"date-time":"2023-01-26T12:31:38Z","timestamp":1674736298553},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"A partition structure is a sequence of probability distributions for $\\pi_n$, a random partition of $n$, such that if $\\pi_n$ is regarded as a random allocation of $n$ unlabeled balls into some random number of unlabeled boxes, and given $\\pi_n$ some $x$ of the $n$ balls are removed by uniform random deletion without replacement, the remaining random partition of $n-x$ is distributed like $\\pi_{n-x}$, for all $1 \\le x \\le n$. We call a partition structure regenerative if for each $n$ it is possible to delete a single box of balls from $\\pi_n$ in such a way that for each $1 \\le x \\le n$, given the deleted box contains $x$ balls, the remaining partition of $n-x$ balls is distributed like $\\pi_{n-x}$. Examples are provided by the Ewens partition structures, which Kingman characterised by regeneration with respect to deletion of the box containing a uniformly selected random ball. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) is associated in turn with a regenerative random subset of the positive halfline. Such a regenerative random set is the closure of the range of a subordinator (that is an increasing process with stationary independent increments). The probability distribution of a general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator, for which exponent an integral representation is provided by the L\u00e9vy-Khintchine formula. The extended Ewens family of partition structures, previously studied by Pitman and Yor, with two parameters $(\\alpha,\\theta)$, is characterised for $0 \\le \\alpha < 1$ and $\\theta >0$ by regeneration with respect to deletion of each distinct part of size $x$ with probability proportional to $(n-x)\\tau+x(1-\\tau)$, where $\\tau = \\alpha\/(\\alpha+\\theta)$.<\/jats:p>","DOI":"10.37236\/1869","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:19:57Z","timestamp":1578719997000},"source":"Crossref","is-referenced-by-count":15,"title":["Regenerative Partition Structures"],"prefix":"10.37236","volume":"11","author":[{"given":"Alexander","family":"Gnedin","sequence":"first","affiliation":[]},{"given":"Jim","family":"Pitman","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,1,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r12\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r12\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:52:20Z","timestamp":1579323140000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i2r12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,1,7]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2004,6,3]]}},"URL":"http:\/\/dx.doi.org\/10.37236\/1869","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,1,7]]}}}