Skip to main content
SpringerLink
Log in

Generalizations of Stanley’s Theorem: Combinatorial Proofs and Related Inequalities

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We give combinatorial proofs for the generalizations of Stanley’s Theorem given in Andrews and Merca (Math Stud 89(1–2): 175–180, 2020). These involve the total number \(b_k(n)\) of parts equal to k in all partitions of n. We also introduce several infinite families of linear inequalities involving \(b_k(n)\).

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

References

  1. Andrews, G.E.: The Theory of Partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1998).. (Reprint of the 1976 original)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Andrews, G.E., Merca, M.: Truncated theta series and a problem of Guo and Zeng. J. Combin. Theory Ser. A 154, 610–619 (2018)

    Article  MathSciNet  Google Scholar 

  4. Andrews, G.E., Merca, M.: A new generalization of Stanley’s theorem. Math. Stud. 89(1–2), 175–180 (2020)

    Google Scholar 

  5. Ballantine, C., Merca, M.: On identities of Watson type. Ars Math. Contemp. 17(1), 277–290 (2019)

    Article  MathSciNet  Google Scholar 

  6. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encycl. Math. Appl. Cambridge University Press, Cambridge (2004)

  7. Merca, M., Wang, C., Yee, A.J.: A truncated theta identity of Gauss and overpartitions into odd parts. Ann. Comb. 23, 907–915 (2019)

    Article  MathSciNet  Google Scholar 

  8. Uncu, A.K.: Weighted Rogers–Ramanujan partitions and Dyson crank. Ramanujan J 46(2), 579–591 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referee for pointing out the refinement of Theorem 1 mentioned at the end of Sect. 2.1, as well as the simpler argument for the proof of Theorem 2 mentioned at the end of Sect. 2.2.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, College of The Holy Cross, Worcester, MA, 01610, USA

    Cristina Ballantine

  2. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Mircea Merca

Authors
  1. Cristina Ballantine
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mircea Merca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristina Ballantine.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ballantine, C., Merca, M. Generalizations of Stanley’s Theorem: Combinatorial Proofs and Related Inequalities. Mediterr. J. Math. 19, 20 (2022). https://doi.org/10.1007/s00009-021-01930-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-021-01930-2

Keywords

Mathematics Subject Classification

Search

Navigation