Skip to main content
SpringerLink
Log in

The First Positive Rank and Crank Moments for Overpartitions

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.

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. Andrews G.E.: Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks. Invent. Math. 169(1), 37–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews G.E.: The number of smallest parts in the partitions of n. J. Reine Angew. Math. 624, 133–142 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Andrews G.E., Chan S.H., Kim B.: The odd moments of ranks and cranks. J. Combin. Theory Ser. A 120(1), 77–91 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrews G.E., Garvan F.G.: Dyson’s crank of a partition. Bull. Amer. Math. Soc. 18(2), 167–171 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Atkin A.O.L., Garvan F.G.: Relations between the ranks and cranks of partitions. Ramanujan J. 7(1-3), 343–366 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bringmann K.: On the explicit construction of higher deformations of partition statistics. Duke Math. J. 144(2), 195–233 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bringmann K., Garvan F., Mahlburg K.: Partition statistics and quasiharmonic Maass forms. Int. Math. Res. Not. 2009(1), 63–97 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Bringmann K., Lovejoy J.: Overpartitions and class numbers of binary quadratic forms. Proc. Natl. Acad. Sci. USA 106(14), 5513–5516 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bringmann K., Lovejoy J., Osburn R.: Rank and crank moments for overpartitions. J. Number Theory 129(7), 1758–1772 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bringmann K., Mahlburg K.: Inequalities between ranks and cranks. Proc. Amer. Math. Soc. 137(8), 2567–2574 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bringmann K., Mahlburg K.: Asymptotic inequalities for positive crank and rank moments. Trans. Amer. Math. Soc. 366(2), 1073–1094 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bringmann K., Mahlburg K., Rhoades R.: Asymptotics for rank and crank moments. Bull. Lond. Math. Soc. 43(4), 661–672 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bringmann K., Mahlburg K., Rhoades R.: Taylor coefficients of mock-Jacobi forms and moments of partition statistics. Math. Proc. Camb. Phil. Soc. 157(2), 231–251 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen W.Y.C., Sang D., Shi Y.: The Rogers-Ramanujan-Gordon theorem for overpartitions. Proc. London. Math. Soc. (3) 106(6), 1371–1393 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Corteel S., Lovejoy J.: Overpartitions. Trans. Amer. Math. Soc. 356(4), 1623–1635 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Desrosiers P., Lapointe L., Mathieu P.: Evaluation and normalization of Jack superpolynomials. Int. Math. Res. Not. 2012(23), 5267–5327 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Diaconis P., Janson S., Rhoades R.: Note on a partition limit theorem for rank and crank. Bull. London Math. Soc. 45(3), 551–553 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dyson F.J.: Some guesses in the theory of partitions. Eureka 8, 10–15 (1944)

    MathSciNet  Google Scholar 

  19. Folsom A., Ono K.: The spt-function of Andrews. Proc. Natl. Acad. Sci. USA 105(51), 20152–20156 (2008)

    Article  MathSciNet  Google Scholar 

  20. Garvan F.G.: Congruences for Andrews’ smallest parts partition function and new congruences for Dyson’s rank. Int. J. Number Theory 6(2), 281–309 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Garvan F.G.: Higher order spt-functions. Adv. Math. 228(1), 241–265 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Garvan F., Jennings-Shaffer C.: The spt-crank for overpartitions. Acta Arith. 166(2), 141–188 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jennings-Shaffer C.: Higher order spt functions for overpartitions, overparitions with smallest part even and partitions without repeated odd parts. J. Number Theory 149, 285–312 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kim B., Kim E., Seo J.: On the number of even and odd strings along the overpartitions of n. Arch. Math. 102(4), 357–368 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Larsen A., Rust A., Swisher H.: Inequalities for positive rank and crank moments of overpartitions. Int. J. Number Theory 10, 2115–2133 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lovejoy J.: Rank and conjugation for the Frobenius representation of an overpartition. Ann. Combin. 9(3), 321–334 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lovejoy J.: Rank and conjugation for a second Frobenius representation of an overpartition. Ann. Combin. 12(1), 101–113 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mao R.: Inequalities between rank moments of overpartitions. J. Number Theory 133(11), 3611–3619 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rhoades R.: Families of quasimodular forms and Jacobi forms: the crank statistic for partitions. Proc. Amer. Math. Soc. 141(1), 29–39 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rolon, J.M.Z.: Asymptotics of higher order spt-functions for overpartitions. Ann. Combin. (to appear)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    George Andrews

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang link, Singapore, 637371, Republic of Singapore

    Song Heng Chan

  3. School of Liberal Arts, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul, 01811, Korea

    Byungchan Kim

  4. School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

    Robert Osburn

Authors
  1. George Andrews
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Song Heng Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Byungchan Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Robert Osburn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Byungchan Kim.

Additional information

The first author was partially supported by National Security Agency Grant H98230-12-1-0205. The second author was partially supported by the Singapore Ministry of Education Academic Research Fund, Tier 1, project number RG68/10. The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF2011-0009199), and the TJ Park Science Fellowship from the POSCO TJ Park Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Andrews, G., Chan, S.H., Kim, B. et al. The First Positive Rank and Crank Moments for Overpartitions. Ann. Comb. 20, 193–207 (2016). https://doi.org/10.1007/s00026-016-0306-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-016-0306-0

Mathematics Subject Classification

Keywords

Search

Navigation