Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Partition identities involving gaps and weights

Author: Krishnaswami Alladi
Journal: Trans. Amer. Math. Soc. 349 (1997), 5001-5019
MSC (1991): Primary 11P83, 11P81; Secondary 05A19
MathSciNet review: 1401759
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers-Ramanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.

References [Enhancements On Off] (What's this?)

  • 1. K. Alladi, A combinatorial correspondence related to Göllnitz' (big) partition theorem and applications, Trans. Amer. Math. Soc. (to appear).
  • 2. K. Alladi, On the divisibility of certain partition functions by small powers of 2 (in preparation).
  • 3. K. Alladi, G. E. Andrews and B. Gordon, Refinements and generalizations of Capparelli's conjecture on partitions, J. Algebra 174 (1995), 636-658. MR 96b:11136
  • 4. K. Alladi and B. Gordon, Partition identities and a continued fraction of Ramanujan, J. Comb. Th. Ser. A 63 (1993), 275-300. MR 94f:11108
  • 5. G. E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley, Reading (1976). MR 58:27738
  • 6. G. E. Andrews, Partitions and Durfee dissection, Amer. J. Math. 10 (1979), 735-742. MR 80h:10020
  • 7. G. E. Andrews, Partitions: Yesterday and Today, New Zealand Math. Soc., Wellington (1979). MR 81b:01015
  • 8. D. Bowman, Partitions with numbers in their gaps, Acta Arith. 74 (1996), 97-105. MR 96m:11090
  • 9. D. M. Bressoud, On a partition theorem of Göllnitz, J. Reine Angew. Math. 305 (1979), 215-217. MR 80a:10027
  • 10. F. Garvan, Generalizations of Dyson's rank and non-Rogers-Ramanujan partitions, Manus. Math 84 (1994), 343-359. MR 95e:11115
  • 11. H. Göllnitz, Partitionen with Differenzenbedingungen, J. Reine Angew. Math 225 (1967), 154-190. MR 35:2848
  • 12. B. Gordon, A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math. 83 (1961), 393-399. MR 23:A809
  • 13. B. Gordon and K. Ono, Divisibility of certain partition functions by powers of primes, The Ramanujan Journal 1 (1997), 25-34.
  • 14. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Clarendon Press, Oxford (1960). MR 81i:10002 (5th ed.)
  • 15. J. P. Serre, Divisibilité des coefficients des formes modulaires de poids entiers, C. R. Acad. Sci. Paris A 279 (1974), 679-682. MR 52:3060
  • 16. J. J. Sylvester, A constructive theory of partitions arranged in three Acts, an Interact and an Exodion, Amer. J. Math. 5 (1882), 251-330.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11P83, 11P81, 05A19

Retrieve articles in all journals with MSC (1991): 11P83, 11P81, 05A19

Additional Information

Krishnaswami Alladi
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611

Keywords: Partitions, weights, gaps, Durfee squares, powers of 2
Received by editor(s): October 10, 1995
Received by editor(s) in revised form: June 3, 1996
Additional Notes: Research supported in part by the National Science Foundation Grant DMS 9400191.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society