Skip to main content
SpringerLink
Log in

Arithmetic of the 7-regular bipartition function modulo 3

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

In this paper, we study the function B ℓ (n) which counts the number of ℓ-regular bipartitions of n. Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). In particular, using Ramanujan’s two modular equations of degree 7, we prove an infinite family of congruences: for α≥2 and n≥0,

$$B_7 \biggl(3^{\alpha}n+\frac{5\cdot 3^{\alpha-1}-1}{2} \biggr)\equiv 0\ ({ \rm mod\ }3). $$

In addition, we give an elementary proof of two infinite families of congruences modulo 3 satisfied by the 7-regular partition function due to Furcy and Penniston (Ramanujan J. 27:101–108, 2012). We also present two conjectures for B 13(n) modulo 3.

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. Ahlgren, S., Lovejoy, J.: The arithmetic of partitions into distinct parts. Mathematika 48, 203–211 (2001)

    Article  MathSciNet  Google Scholar 

  2. Andrews, G.E., Hirschhorn, M.D., Sellers, J.A.: Arithmetic properties of partitions with even parts distinct. Ramanujan J. 23, 169–181 (2010)

    Article  MathSciNet  Google Scholar 

  3. Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)

    Book  Google Scholar 

  4. Calkin, N., Drake, N., James, K., Law, S., Lee, P., Penniston, D., Radder, J.: Divisibility properties of the 5-regular and 13-regular partition functions. Integers 8, A60 (2008)

    MathSciNet  Google Scholar 

  5. Chen, S.C.: On the number of partitions with distinct even parts. Discrete Math. 311, 940–943 (2011)

    Article  MathSciNet  Google Scholar 

  6. Chen, S.C.: Congruences for the number of k-tuple partitions with distinct even parts. Discrete Math. 313, 1565–1568 (2013)

    Article  MathSciNet  Google Scholar 

  7. Cui, S.P., Gu, N.S.S.: Arithmetic properties of the ℓ-regular partitions. Adv. Appl. Math. 51, 507–523 (2013)

    Article  MathSciNet  Google Scholar 

  8. Dandurand, B., Penniston, D.: ℓ-divisibility of ℓ-regular partition functions. Ramanujan J. 19, 63–70 (2009)

    Article  MathSciNet  Google Scholar 

  9. Furcy, D., Penniston, D.: Congruences for ℓ-regular partition functions modulo 3. Ramanujan J. 27, 101–108 (2012)

    Article  MathSciNet  Google Scholar 

  10. Gordon, B., Ono, K.: Divisibility of certain partition functions by powers of primes. Ramanujan J. 1, 25–34 (1997)

    Article  MathSciNet  Google Scholar 

  11. Hirschhorn, M.D., Sellers, J.A.: Elementary proofs of parity results for 5-regular partitions. Bull. Aust. Math. Soc. 81, 58–63 (2010)

    Article  MathSciNet  Google Scholar 

  12. Lin, B.L.S.: Arithmetic properties of bipartitions with even parts distinct. Ramanujan J. (2013). doi:10.1007/s11139-013-9473-3

    Google Scholar 

  13. Lovejoy, J.: The divisibility and distribution of partitions into distinct parts. Adv. Math. 158, 253–263 (2001)

    Article  MathSciNet  Google Scholar 

  14. Lovejoy, J.: The number of partitions into distinct parts modulo powers of 5. Bull. Lond. Math. Soc. 35, 41–46 (2003)

    Article  MathSciNet  Google Scholar 

  15. Lovejoy, J., Penniston, D.: 3-regular partitions and a modular K3 surface. Contemp. Math. 291, 177–182 (2001)

    Article  MathSciNet  Google Scholar 

  16. Ono, K., Penniston, D.: The 2-adic behavior of the number of partitions into distinct parts. J. Comb. Theory, Ser. A 92, 138–157 (2000)

    Article  MathSciNet  Google Scholar 

  17. Penniston, D.: The p a-regular partition function modulo p j. J. Number Theory 94, 320–325 (2002)

    Article  MathSciNet  Google Scholar 

  18. Penniston, D.: Arithmetic of ℓ-regular partition functions. Int. J. Number Theory 4, 295–302 (2008)

    Article  MathSciNet  Google Scholar 

  19. Webb, J.J.: Arithmetic of the 13-regular partition function modulo 3. Ramanujan J. 25, 49–56 (2011)

    Article  MathSciNet  Google Scholar 

  20. Xia, E.X.W., Yao, O.X.M.: Parity results for 9-regular partitions. Ramanujan J. (2013). doi:10.1007/s11139-013-9493-z

    Google Scholar 

Download references

Acknowledgements

The author would like to thank the referee for helpful suggestions.

Author information

Authors and Affiliations

  1. School of Sciences, Jimei University, Xiamen, 361021, China

    Bernard L. S. Lin

Authors
  1. Bernard L. S. Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernard L. S. Lin.

Additional information

This work was supported by the National Science Foundation of China (Tianyuan Fund for Mathematics, No. 11226299), and the Natural Science Foundation of Fujian Province of China (No. 2013J05011).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, B.L.S. Arithmetic of the 7-regular bipartition function modulo 3. Ramanujan J 37, 469–478 (2015). https://doi.org/10.1007/s11139-013-9542-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-013-9542-7

Keywords

Mathematics Subject Classification

Search

Navigation