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The Size of the Largest Part of Random Weighted Partitions of Large Integers

Published online by Cambridge University Press:  21 February 2013

LJUBEN MUTAFCHIEV
LJUBEN MUTAFCHIEV*
Affiliation:
American University in Bulgaria, 2700 Blagoevgrad, Bulgaria and Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences (e-mail: ljuben@aubg.bg)
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Abstract

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We consider partitions of the positive integer n whose parts satisfy the following condition. For a given sequence of non-negative numbers {bk}k≥1, a part of size k appears in exactly bk 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 Xn. Let D(s)=∑k=1∞bkk−s, s=σ+iy, be the Dirichlet generating series of the weights bk. 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 Xn, 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.

Keywords

Primary 05A17Secondary 60C0560F05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 3 , May 2013 , pp. 433 - 454
Copyright
Copyright © Cambridge University Press 2013

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