Skip to main content
SpringerLink
Log in

Dyson’s partition ranks and their multiplicative extensions

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We study the Dyson rank function N(r, 3; n), the number of partitions of n with rank \(\equiv r \pmod 3\). We investigate the convexity of these functions. We extend N(r, 3; n) multiplicatively to the set of partitions, and we determine the maximum value when taken over all partitions of size n.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Dyson’s rank does not explain Ramanujan’s congruence modulo 11.

  2. This follows immediately from considering conjugations of Ferrers diagrams.

References

  1. Apostol, T.M.: Modular functions and Dirichlet series in number theory. Grad. Texts Math. 10, 94–112 (1976)

    Article  Google Scholar 

  2. Atkin, A.L., Swinnerton-Dyer, P.: Modular forms on noncongruence subgroups. Lond. Math. Soc. 3, 84–106 (1954)

    Article  MATH  Google Scholar 

  3. Bessenrodt, C., Ono, K.: Maximal multiplicative properties of partitions. Ann. Comb. 20, 59–64 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bringmann, K.: Asymptotics for rank partition functions. Trans. Am. Math. Soc. 361, 3483–3500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bringmann, K., Ono, K.: The \(f(q)\) mock theta function conjecture and partition ranks. Invent. Math. 165, 243–266 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dyson, E.: Some guesses in the theory of partitions. Eureka 8, 10–15 (1944)

    MathSciNet  Google Scholar 

  7. Lehmer, D.H.: On the remainders and convergence of the series for the partition function. Trans. Am. Math. Soc. 46, 362–373 (1939)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ramanujan, S.: Congruence properties of partitions. Math. Z. 9, 147–153 (1919)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the NSF and the Emory REU (especially Professor Ono) for their support of our research.

Author information

Authors and Affiliations

  1. Department of Mathematics, Yale University, New Haven, CT, 06520, USA

    Elaine Hou

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Meena Jagadeesan

Authors
  1. Elaine Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Meena Jagadeesan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elaine Hou.

Additional information

Funding was provided by National Science Foundation (Grant Nos. DMS-1557960 and DMS-1557960).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hou, E., Jagadeesan, M. Dyson’s partition ranks and their multiplicative extensions. Ramanujan J 45, 817–839 (2018). https://doi.org/10.1007/s11139-016-9881-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-016-9881-2

Keywords

Mathematics Subject Classification

Search

Navigation