Advertisement

The Ramanujan Journal

, Volume 46, Issue 2, pp 579–591 | Cite as

Weighted Rogers–Ramanujan partitions and Dyson crank

  • Ali Kemal Uncu
Article
  • 120 Downloads

Abstract

In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts’ sum is n is equal to the number of partitions of n with non-negative crank.

Keywords

Dyson crank Partitions Rogers–Ramanujan Weighted partition identities 

Mathematics Subject Classification

05A15 05A17 05A19 11B75 11P81 11P84 

Notes

Acknowledgements

The author would like to thank George E. Andrews and Alexander Berkovich for their guidance. We would like to express gratitude to the anonymous referee for the constructive comments and making this work more presentable. The author would also like to thank Alexander Berkovich, Jeramiah Hocutt, Frank Patane, and John Pfeilsticker for their helpful comments on the manuscript.

References

  1. 1.
    Alladi, K.: Partition identities involving gaps and weights. Trans. Am. Math. Soc. 349(12), 5001–5019 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alladi, K., Berkovich, A.: Göllnitz–Gordon Partitions with Weights and Parity Conditions. Zeta Functions, Topology and Quantum Physics. Developments in Mathematics, vol. 14. Springer, New York (2005)MATHGoogle Scholar
  3. 3.
    Alladi, K., Berkovich, A.: New weighted Rogers–Ramanujan partition theorems and their implications. Trans. Am. Math. Soc. 354(7), 2557–2577 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Andrews, G.E.: The Theory of Partitions. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1998). Reprint of the 1976 original. MR1634067 (99c:11126)Google Scholar
  5. 5.
    Andrews, G.E., Garvan, F.G.: Dyson’s Crank of a Partition. Bull. Am. Math. Soc. (N.S.) 18(2), 167–171 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Auluck, F.C.: On some new types of partitions associated with generalized Ferrers graphs. Proc. Camb. Philos. 47, 679–686 (1951)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Berkovich, A., Garvan, F.G.: Some observations on Dysons new symmetries of partitions. J. Comb. Theory A 100(1), 61–93 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Berkovich, A., Uncu, A.K.: On partitions with fixed number of even-indexed and odd-indexed odd parts. J. Number Theory 167, 7–30 (2016)Google Scholar
  9. 9.
    Boulet, C.E.: A four-parameter partition identity. Ramanujan J. 12(3), 315–320 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ishikawa, M., Zeng, J.: The Andrews–Stanley partition function and Al-Salam–Chihara polynomials. Discrete Math. 309(1), 151–175 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

Personalised recommendations