Transactions of the American Mathematical Society

Author(s): Sylvie Corteel; Jeremy Lovejoy
Journal: Trans. Amer. Math. Soc. 356 (2004), 1623-1635.
MSC (2000): Primary 11P81, 05A17, 33D15
Posted: July 8, 2003
Retrieve article in: PDF DVI PostScript

Abstract: We discuss a generalization of partitions, called overpartitions, which have proven useful in several combinatorial studies of basic hypergeometric series. After showing how a number of finite products occurring in

-series have natural interpretations in terms of overpartitions, we present an introduction to their rich structure as revealed by

-series identities.

1.: K. Alladi, A fundamental invariant in the theory of partitions, Topics in Number Theory, Math. Appl. 467, Kluwer, Dordrecht, 1999. MR 2000e:05011
2.: G. E. Andrews, On a calculus of partition functions, Pacific J. Math. 31 (1969), 555-562. MR 40:7128
3.: G .E. Andrews, Generalizations of the Durfee square, J. London Math. Soc. 3 (1971), 563-570. MR 45:80
4.: G. E. Andrews, Problems and prospects for basic hypergeometric functions, Theory and Application of Special Functions, Academic Press, New York, 1975. MR 53:3372
5.: G. E. Andrews, Partitions: Yesterday and Today, New Zealand Mathematical Society, Wellington, New Zealand, 1979. MR 81b:01015
6.: G. E. Andrews, Partitions and Durfee dissection, Amer. J. Math. 101 (1979), 735-742. MR 80h:10020
7.: G. E. Andrews, Ramanujan's ``lost'' notebook I. Partial $\theta$ functions, Adv. Math. 41 (1981), 137-172. MR 83m:10034a
8.: G. E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998. MR 99c:11126
9.: G. E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc. 49 (1984), no. 301. MR 85m:11063
10.: G. E. Andrews, F. J. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), 391-407. MR 89f:11071
11.: G. E. Andrews, J. Jimenez-Urroz, and K. Ono, -series identities and values of certain -functions, Duke Math. J. 108 (2001), 395-419. MR 2002e:11055
12.: D. Bressoud, and M. Subbarao, On Uchimura's connection between partitions and the number of divisors, Canad. Math. Bull. 27 (1984), no. 2, 143-145. MR 85i:11087
13.: R. Chapman, Partition identities arising from involutions, Australas. J. Combin. 27 (2003), 285-291.
14.: S. Corteel, Particle seas and basic hypergeometric series, Adv. Appl. Math., to appear.
15.: S. Corteel and J. Lovejoy, Frobenius partitions and the combinatorics of Ramanujan's $_1 \psi _1$ summation, J. Combin. Theory Ser. A 97 (2002), 177-183. MR 2002k:11181
16.: N. J. Fine, Basic Hypergeometric Series and Applications, American Mathematical Society, Providence, RI, 1988. MR 91j:33011
17.: G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990. MR 91d:33034
18.: J. T. Joichi and D. Stanton, Bijective proofs of basic hypergeometric series identities, Pacific J. Math. 127 (1987), 103-120. MR 88h:33004
19.: J. Lovejoy, A Bailey lattice, Proc. Amer. Math. Soc., to appear.
20.: I. Pak, Partition bijections: A survey, preprint.
21.: K. Uchimura, An identity for the divisor generating function arising from sorting theory, J. Combin. Theory Ser. A 31 (1981), 131-135. MR 82k:05015
22.: S. O. Warnaar, Partial theta functions I. Beyond the lost notebook, preprint.
23.: J. Zeng, The -variations of Sylvester's bijection between odd and strict partitions, preprint.

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11P81, 05A17, 33D15

Sylvie Corteel
Affiliation: CNRS, PRiSM, UVSQ, 45 Avenue des Etats Unis, 78035 Versailles Cedex, France
Email: Sylvie.Corteel@prism.uvsq.fr

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS