The Ramanujan Journal

, Volume 34, Issue 1, pp 109–117 | Cite as

Parity results for 9-regular partitions

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


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


Partition Regular partition Congruence 

Mathematics Subject Classification

11P83 05A17 



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.


  1. 1.
    Ahlgren, S., Lovejoy, J.: The arithmetic of partitions into distinct parts. Mathematika 48, 203–211 (2001) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Alladi, K.: Partition identities involving gaps and weights. Trans. Am. Math. Soc. 349, 5001–5019 (1997) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Andrews, G.E.: Euler’s “De Partitio Numerorum”. Bull. Am. Math. Soc. 44, 561–573 (2007) CrossRefMATHGoogle Scholar
  4. 4.
    Andrews, G.E., Hirschhorn, M.D., Sellers, J.A.: Arithmetic properties of partitions with even parts distinct. Ramanujan J. 23, 169–181 (2010) CrossRefMATHMathSciNetGoogle Scholar
  5. 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) CrossRefMATHMathSciNetGoogle Scholar
  6. 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) MathSciNetGoogle Scholar
  7. 7.
    Dandurand, B., Penniston, D.: l-Divisiblity of l-regular partition functions. Ramanujan J. 19, 63–70 (2009) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Furcy, D., Penniston, D.: Congruences for l-regular partition functions modulo 3. Ramanujan J. 27, 101–108 (2012) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Gordon, B., Ono, K.: Divisibility of certain partition functions by powers of primes. Ramanujan J. 1, 25–34 (1997) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hirschhorn, M.D., Sellers, J.A.: Elementary proofs of parity results for 5-regular partitions. Bull. Aust. Math. Soc. 81, 58–63 (2010) CrossRefMATHMathSciNetGoogle Scholar
  11. 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) CrossRefGoogle Scholar
  12. 12.
    Penniston, D.: The p a-regular partition function modulo p j. J. Number Theory 94, 320–325 (2002) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Penniston, D.: Arithmetic of l-regular partition functions. Int. J. Number Theory 4, 295–302 (2008) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Treneer, S.: Quadratic twists and the coefficients of weakly holomorphic modular forms. J. Ramanujan Math. Soc. 23, 283–309 (2008) MATHMathSciNetGoogle Scholar
  15. 15.
    Webb, J.J.: Arithmetic of the 13-regular partition function modulo 3. Ramanujan J. 25, 49–56 (2011) CrossRefMATHMathSciNetGoogle Scholar
  16. 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) CrossRefMATHMathSciNetGoogle Scholar
  17. 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

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsJiangsu UniversityZhenjiangP.R. China

Personalised recommendations