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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

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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References

[1]Abramovitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover.Google Scholar
[2]Andrews, G. E. (1976) The Theory of Partitions, Vol. 2 of Encyclopedia of Mathematics and its Applications, Addison-Wesley.Google Scholar
[3]Arratia, R., Barbour, A. D. and Tavaré, S. (2003) Logarithmic Combinatorial Structures: A Probabilistic Approach, European Mathematical Society.Google Scholar
[4]Auluck, F. C., Chowla, S. and Gupta, H. (1942) On the maximum value of the number of partitions of n into k parts. J. Indian Math. Soc. 6 105–112.Google Scholar
[5]Bateman, H. and Erdélyi, A. (1954) Tables of Integral Transforms, McGraw-Hill.Google Scholar
[6]Erdős, P. and Lehner, J. (1941) The distribution of the number of summands in the partition of a positive integer. Duke Math. J. 8 335–345.Google Scholar
[7]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[8]Freiman, G. A. and Granovsky, B. L. (2002) Asymptotic formula for a partition function of reversible coagulation–fragmentation processes. Israel J. Math. 130 259–279.Google Scholar
[9]Fristedt, B. (1993) The structure of random partitions of large integers. Trans. Amer. Math. Soc. 337 703–735.CrossRefGoogle Scholar
[10]Granovsky, B., Stark, D. and Erlihson, M. (2008) Meinardus theorem on weighted partitions: Extensions and a probabilistic proof. Adv. Appl. Math. 41 307–328.Google Scholar
[11]Hayman, W. K. (1956) A generalization of Stirling's formula. J. Reine Angew. Math. 196 67–95.Google Scholar
[12]Korevaar, J. (2004) Tauberian Theory, Springer.Google Scholar
[13]Meinardus, G. (1954) Asymptotische Aussagen über Partitionen. Math. Z. 59 388–398.Google Scholar
[14]Mutafchiev, L. (2006) The size of the largest part of random plane partitions of large integers. Integers: Electron. J. Combin. Number Theory 6 #A13.Google Scholar
[15]Mutafchiev, L. (2011) Limit theorems for the number of parts in a random weighted partition. Electron. J. Combin. 18 #P206.Google Scholar
[16]Nijenhuis, A. and Wilf, H. (1978) Combinatorial Algorithms, second edition, Academic Press.Google Scholar
[17]Pittel, B. (2005) On dimensions of a random solid diagram. Combin. Probab. Comput. 14 873–895.Google Scholar
[18]Prachar, K. (1957) Primzahlverteilung, Springer.Google Scholar
[19]Stanley, R. P. (1999) Enumerative Combinatorics 2, Vol. 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
[20]Titchmarsh, E. C. (1939) The Theory of Functions, Oxford University Press.Google Scholar
[21]Vershik, A. (1996) Statistical mechanics of combinatorial partitions and their limit configurations. Funct. Anal. Appl. 30 90–105.CrossRefGoogle Scholar
[22]Vershik, A. and Yakubovich, Y. (2006) Fluctuations of the maximal particle energy of the quantum ideal gas and random partitions. Comm. Math. Phys. 261 759–769.Google Scholar
[23]Whittaker, E. T. and Watson, G. N. (1927) A Course of Modern Analysis, Cambridge University Press.Google Scholar
[24]Wilf, H. (1994) generatingfunctionology, Academic Press.Google Scholar