Skip to main content
SpringerLink
Log in

Parity results for 9-regular partitions

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Let t≥2 be an integer. We say that a partition is t-regular if none of its parts is divisible by t, and denote the number of t-regular partitions of n by b t (n). In this paper, we establish several infinite families of congruences modulo 2 for b 9(n). For example, we find that for all integers n≥0 and k≥0,

$$b_9 \biggl(2^{6k+7}n+ \frac{2^{6k+6}-1}{3} \biggr)\equiv 0 \quad (\mathrm{mod}\ 2 ). $$

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.

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  MATH  MathSciNet  Google Scholar 

  2. Alladi, K.: Partition identities involving gaps and weights. Trans. Am. Math. Soc. 349, 5001–5019 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andrews, G.E.: Euler’s “De Partitio Numerorum”. Bull. Am. Math. Soc. 44, 561–573 (2007)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Baruah, N.D., Ojah, K.K.: Analogues of Ramanujan’s partition identities and congruences arising from his theta functions and modular equations. Ramanujan J. 28, 385–407 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Lovejoy, J., Penniston, D.: 3-regular partitions and a modular K3 surface. In: q-Series with Applications to Combinatorics, Number Theory, and Physics, Urbana, IL, 2000. Contemp. Math., vol. 291, pp. 177–182. Amer. Math. Soc., Providence (2001)

    Chapter  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Treneer, S.: Quadratic twists and the coefficients of weakly holomorphic modular forms. J. Ramanujan Math. Soc. 23, 283–309 (2008)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Xia, E.X.W., Yao, O.X.M.: Some modular relations for the Göllnitz—Gordon functions by an even—odd method. J. Math. Anal. Appl. 387, 126–138 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Xia, E.X.W., Yao, O.X.M.: Analogues of Ramanujan’s partition identities. Ramanujan J., in press. doi:10.1007/s11139-012-9439-x

Download references

Acknowledgements

The authors would like to thank the anonymous referees for valuable suggestions, corrections, and comments that resulted in a great improvement of the original manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu, 212013, P.R. China

    Ernest X. W. Xia & Olivia X. M. Yao

Authors
  1. Ernest X. W. Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Olivia X. M. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olivia X. M. Yao.

Additional information

This work was supported by the National Science Foundation of China and PAPD.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xia, E.X.W., Yao, O.X.M. Parity results for 9-regular partitions. Ramanujan J 34, 109–117 (2014). https://doi.org/10.1007/s11139-013-9493-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-013-9493-z

Keywords

Mathematics Subject Classification

Search

Navigation