Skip to main content
SpringerLink
Log in

General Colored Partition Identities

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

Ramanujan’s modular equations of degrees 3, 5, 7, 11 and 23 yield beautiful colored partition identities. Warnaar analytically generalized the modular equations of degrees 3 and 7,  and thereafter, the author found bijective proofs of those partitions identities and recently, established an analytic generalization of the modular equations of degrees 5, 11 and 23. The partition identities of degrees 5 and 11 were combinatorially proved by Sandon and Zanello, and it remains open to find a combinatorial proof of the partition identity of degree 23. In this paper, we prove general colored partition identities with a restriction on the number of parts, which are connected to the partition identities arising from those modular equations. We also provide bijective proofs of these partition identities. In particular, one of these proofs gives bijective proofs of the partition identity of degree 23 for some cases, which also work for the identities of degrees 5 and 11 for the same cases.

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

References

  1. N. D. Baruah and B. C. Berndt, Partition identities and Ramanujan’s modular equations, J. Comb. Thy. (A) 114, 1024–1045 (2007).

    Article  MathSciNet  Google Scholar 

  2. B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.

    Book  Google Scholar 

  3. B. C. Berndt, Partition-theoretic interpretations of certain modular equations of Schröter, Russell, and Ramanujan. Ann. Combin. 11 (2), 115–125 (2007).

    Article  MathSciNet  Google Scholar 

  4. B. C. Berndt and R. R. Zhou, Proofs of conjectures of Sandon and Zanello on colored partition identities, J. Korean Math. Soc., 51 No. 5, 987–1028 (2014).

    Article  MathSciNet  Google Scholar 

  5. B. C. Berndt and R. R. Zhou, Identities for partitions with distinct colors, Ann. Combin., 19 397–420 (2015).

    Article  MathSciNet  Google Scholar 

  6. H. M. Farkas and I. Kra, Partitions and theta constant identities, in: Analysis, Geometry, Number Theory: the Mathematics of Leon Ehrenpreis, Contemp. Math. 251, 197–203 (2000).

  7. H. M. Farkas and I. Kra, Theta Constants, Riemann Surfaces and the Modular Group, Graduate Studies in Mathematics, Vol. 37, American Mathematical Society, Providence, RI, 2001.

    Google Scholar 

  8. G. Gasper and M. Rahman, Basic Hypergeomertic Series, Second Edition, Cambridge University Press, Cambridge, 2004.

    Book  Google Scholar 

  9. C. Guetzlaff, Aequatio modularis pro transformatione functionum ellipticarum septimi ordinis, J. Reine Angew. Math., 12, 173–177 (1834).

    MathSciNet  Google Scholar 

  10. M. D. Hirschhorn, The case of the mysterious sevens, Internat. J. Number Thy., 2, 213–216 (2006).

    Article  MathSciNet  Google Scholar 

  11. S. Kim, Bijective proofs of partition identities arising from modular equations, J. Combin. Thy. Ser. A, 116 (3), 699–712 (2009).

    Article  MathSciNet  Google Scholar 

  12. S. Kim, A generalization of the modular equations of higher degrees, preprint.

  13. C. Sandon and F. Zanello, Warnaar’s bijection and colored partition identities, I, J. Combin. Thy. Ser. A, 120 (1), 28–38 (2013).

    Article  MathSciNet  Google Scholar 

  14. C. Sandon and F. Zanello, Warnaar’s bijection and colored partition identities, II, Ramanujan Journal, 33 (1), 83–120 (2014).

    Article  MathSciNet  Google Scholar 

  15. S. O. Warnaar, A generalization of the Farkas and Kra partition theorem for modulus 7, J. Combin. Thy. Ser. (A) 110 (2005), 43–52.

    Article  MathSciNet  Google Scholar 

  16. E. M. Wright, An enumerative proof of an identity of Jacobi, J. London Math. Soc. 40 (1965), 55–57.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is very grateful to the referees for their valuable suggestions. The author would like to thank B. C. Berndt and S. O. Warnaar for their helpful comments.

Author information

Authors and Affiliations

  1. AORC, SungKyunKwan University, Suwon, South Korea

    Sun Kim

Authors
  1. Sun Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sun Kim.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of the author was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant agreement no. 335220 - AQSER

This work was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2016R1A5A1008055).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, S. General Colored Partition Identities. Ann. Comb. 24, 425–438 (2020). https://doi.org/10.1007/s00026-020-00497-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-020-00497-1

Keywords

Mathematics Subject Classification

Search

Navigation