<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume8 Issue3" %>
Multiplicity and Number of Parts in Overpartitions
Sylvie Corteel1 and Pawel Hitczenko2
1Chargée de Recherche CNRS affectée au PRiSM, Université de Versailles, Versailles, France
syl@prism.uvsq.fr
2Department of Mathematics and Computer Science, Drexel University, Philadelpia, USA
phitczenko@mcs.drexel.edu
Annals of Combinatorics 8 (3) p.287-301 September, 2004
AMS Subject Classification:05A17, 60C05, 11P82
Abstract:
The purpose of this paper is to study the parts, part sizes and multiplicities in overpartitions using combinatorics, probabilities and asymptotics. We show that the probability that a randomly chosen part size of a randomly chosen overpartition of n has multiplicity m or m+1 approaches 1/(m(m+1)ln 2) and that the expected multiplicity of a randomly chosen part size of a randomly chosen overpartition of n approaches lnn/(4ln2) as n
Keywords: partitions, combinatorial probability

References:

1. G.E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge-New York, 1984.

2. L. Bégin, J.F. Fortin, P. Jacob, and P. Mathieu, Fermionic characters for graded parafermions, Nuclear Phys. B 659 (3) (2003) 365--386.

3. C. Bessenrodt and I. Pak, Partition congruences by involutions, Europ. J. Combin, to appear.

4. S. Corteel, Particle seas and basic hypergeometric series, Adv. Appl. Math. 31 (1) (2003) 199--214.

5. E.R. Canfield, S. Corteel, and P. Hitczenko, Partitions with rth difference non-negative, Adv. Appl. Math. 27 (2001) 298--317.

6. S. Corteel and J. Lovejoy, Frobenius partitions and the combinatorics of Ramanujan's 1Ψ1 summation, J. Combin. Theory Ser. A 97 (2002) 179--183.

7. S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004) 1623--1635.

8. S. Corteel, J. Lovejoy, and A.J. Yee, Overpartitions and generating functions for generalized Frobenius partitions, Trends Math., to appear.

9. S. Corteel, B. Pittel, C.D. Savage, and H.S. Wilf, On the multiplicity of parts in a random partition, Random Structures Algorithms 14 (1999) 185--197.

10. P. Erdös and J. Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8 (1941) 335--345.

11. J.F. Fortin, P. Jacob, and P. Mathieu, Jagged partitions, preprint, arXiv:math.CO/0310079.

12. B. Fristedt, The structure of random partitions of large integers, Trans. Amer. Math. Soc. 337 (1993) 703--735.

13. W.M.Y. Goh and E. Schmutz, The number of distinct part sizes in a random integer partition, J. Combin. Theory Ser. A 69 (1995) 149--158.

14. G.H. Hardy, Ramanujan, Cambridge University Press, Cambridge-New York, 1940.

15. G.H. Hardy and Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. (2) XVII (1918) 75--115.

16. K. Husimi, Partition numerorum as occurring in a problem of nuclear physics, Proceedings of the Physico-Mathematical Society of Japan 20 (1938) 912--925.

17. J.T. Joichi and D. Stanton, Bijective proofs of basic hypergeometric series identities, Pacific J. Math. 127 (1) (1987) 103--120.

18. I. Kessler and M. Livingston, The expected number of parts in a partition of n, Monatsh. Math. 81 (1976) 203--212.

19. A. Knopfmacher and M.E. Mays, The sum of distinct parts in compositions and partitions, Bull. Inst. Combin. Appl. 25 (1999) 66--78.

20. J. Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory Ser. A 103 (2003) 393--401.

21. J. Lovejoy, Overpartitions and real quadratic fields, J. Number Theory 106 (2004) 178--186.

22. J. Lovejoy, Overpartition theorems of the Rogers-Ramanujan type, J. London Math. Soc. (2) 69 (2004) 562--574.

23. H. Rademacher, Topics in Analytic Number Theory, Springer Verlag, Berlin, 1973.

24. A.J. Yee, Combinatorial proofs of Ramanujan's 1Ψ1 summation and the q-Gauss summation, J. Combin. Theory Ser. A 105 (2004) 63--77.