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

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 parti...

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Bibliographic Details
Published in:Combinatorics, probability & computing probability & computing, 2013-05, Vol.22 (3), p.433-454
Main Author: MUTAFCHIEV, LJUBEN
Format: Article
Language:English
Subjects:
Asymptotic properties
Combinatorial analysis
Computation
Dirichlet problem
Distribution functions
Integers
Partitions
Random variables
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Summary: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, −∞
ISSN:0963-5483
1469-2163
DOI:10.1017/S0963548313000047