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- W2154171375 abstract "We consider partitions of the positive integer n whose parts satisfy the following condition. For a given sequence of non-negative numbers { b k } k ≥1 , a part of size k appears in exactly b k 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 X n . Let D(s) =∑ k =1 ∞ b k k −s , s =σ+ iy , be the Dirichlet generating series of the weights b k . Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions as n →∞. Using the Meinardus scheme of conditions, we prove that X n , appropriately normalized, converges weakly to a random variable having Gumbel distribution ( i.e ., its distribution function equals e −e −t , −∞< t <∞). This limit theorem extends some known results on particular types of partitions and on the Bose–Einstein model of ideal gas." @default.
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- W2154171375 date "2013-02-21" @default.
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- W2154171375 title "The Size of the Largest Part of Random Weighted Partitions of Large Integers" @default.
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- W2154171375 doi "https://doi.org/10.1017/s0963548313000047" @default.
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