{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T14:39:15Z","timestamp":1775054355510,"version":"3.50.1"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2013,2,21]],"date-time":"2013-02-21T00:00:00Z","timestamp":1361404800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2013,5]]},"abstract":"<jats:p>We consider partitions of the positive integer<jats:italic>n<\/jats:italic>whose parts satisfy the following condition. For a given sequence of non-negative numbers {<jats:italic>b<jats:sub>k<\/jats:sub><\/jats:italic>}<jats:sub><jats:italic>k<\/jats:italic>\u22651<\/jats:sub>, a part of size<jats:italic>k<\/jats:italic>appears in exactly<jats:italic>b<jats:sub>k<\/jats:sub><\/jats:italic>possible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest part<jats:italic>X<jats:sub>n<\/jats:sub><\/jats:italic>. Let<jats:italic>D(s)<\/jats:italic>=\u2211<jats:sub><jats:italic>k<\/jats:italic>=1<\/jats:sub><jats:sup>\u221e<\/jats:sup><jats:italic>b<jats:sub>k<\/jats:sub>k<jats:sub>\u2212s<\/jats:sub><\/jats:italic>,<jats:italic>s<\/jats:italic>=\u03c3+<jats:italic>iy<\/jats:italic>, be the Dirichlet generating series of the weights<jats:italic>b<jats:sub>k<\/jats:sub><\/jats:italic>. Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions as<jats:italic>n<\/jats:italic>\u2192\u221e. Using the Meinardus scheme of conditions, we prove that<jats:italic>X<jats:sub>n<\/jats:sub><\/jats:italic>, appropriately normalized, converges weakly to a random variable having Gumbel distribution (<jats:italic>i.e<\/jats:italic>., its distribution function equals<jats:italic>e<jats:sup>\u2212e<\/jats:sup><jats:sup>\u2212t<\/jats:sup><\/jats:italic>, \u2212\u221e&lt;<jats:italic>t<\/jats:italic>&lt;\u221e). This limit theorem extends some known results on particular types of partitions and on the Bose\u2013Einstein model of ideal gas.<\/jats:p>","DOI":"10.1017\/s0963548313000047","type":"journal-article","created":{"date-parts":[[2013,2,21]],"date-time":"2013-02-21T13:58:20Z","timestamp":1361455100000},"page":"433-454","source":"Crossref","is-referenced-by-count":6,"title":["The Size of the Largest Part of Random Weighted Partitions of Large Integers"],"prefix":"10.1017","volume":"22","author":[{"given":"LJUBEN","family":"MUTAFCHIEV","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2013,2,21]]},"reference":[{"key":"S0963548313000047_ref23","volume-title":"A Course of Modern Analysis","author":"Whittaker","year":"1927"},{"key":"S0963548313000047_ref3","doi-asserted-by":"publisher","DOI":"10.4171\/000"},{"key":"S0963548313000047_ref17","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548305006917"},{"key":"S0963548313000047_ref24","volume-title":"generatingfunctionology","author":"Wilf","year":"1994"},{"key":"S0963548313000047_ref19","volume-title":"Cambridge Studies in Advanced Mathematics","author":"Stanley","year":"1999"},{"key":"S0963548313000047_ref1","volume-title":"Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables","author":"Abramovitz","year":"1965"},{"key":"S0963548313000047_ref20","volume-title":"The Theory of Functions","author":"Titchmarsh","year":"1939"},{"key":"S0963548313000047_ref11","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1515\/crll.1956.196.67","article-title":"A generalization of Stirling's formula.","volume":"196","author":"Hayman","year":"1956","journal-title":"J. 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Indian Math. Soc."},{"key":"S0963548313000047_ref9","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1993-1094553-1"},{"key":"S0963548313000047_ref6","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1215\/S0012-7094-41-00826-8","article-title":"The distribution of the number of summands in the partition of a positive integer.","volume":"8","author":"Erd\u0151s","year":"1941","journal-title":"Duke Math. J."},{"key":"S0963548313000047_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801655"},{"key":"S0963548313000047_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02764079"},{"key":"S0963548313000047_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-10225-1"},{"key":"S0963548313000047_ref14","first-page":"#A13","article-title":"The size of the largest part of random plane partitions of large integers","volume":"6","author":"Mutafchiev","year":"2006","journal-title":"Integers: Electron. J. Combin. Number Theory"},{"key":"S0963548313000047_ref15","doi-asserted-by":"crossref","first-page":"#P206","DOI":"10.37236\/693","article-title":"Limit theorems for the number of parts in a random weighted partition","volume":"18","author":"Mutafchiev","year":"2011","journal-title":"Electron. J. Combin."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548313000047","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,29]],"date-time":"2023-06-29T18:04:05Z","timestamp":1688061845000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548313000047\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2,21]]},"references-count":24,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,5]]}},"alternative-id":["S0963548313000047"],"URL":"https:\/\/doi.org\/10.1017\/s0963548313000047","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,2,21]]}}}