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Congruences modulo powers of 3 for 2-color partition triples

  • Dazhao Tang
Article
  • 9 Downloads

Abstract

Let \(p_{k,3}(n)\) enumerate the number of 2-color partition triples of n where one of the colors appears only in parts that are multiples of k. In this paper, we prove several infinite families of congruences modulo powers of 3 for \(p_{k,3}(n)\) with \(k=1, 3\), and 9. For example, for all integers \(n\ge 0\) and \(\alpha \ge 1\), we prove that
$$\begin{aligned} p_{3,3}\left( 3^{\alpha }n+\dfrac{3^{\alpha }+1}{2}\right)&\equiv 0\pmod {3^{\alpha +1}} \end{aligned}$$
and
$$\begin{aligned} p_{3,3}\left( 3^{\alpha +1}n+\dfrac{5\times 3^{\alpha }+1}{2}\right)&\equiv 0\pmod {3^{\alpha +4}}. \end{aligned}$$

Keywords

Partition Congruences 2-Color partition triples 

Mathematics Subject Classification

05A17 11P83 

Notes

Acknowledgements

I am indebted to Shishuo Fu for his helpful comments and suggestions that have improved this paper to a great extent. I would like to acknowledge the referee for his/her careful reading and helpful comments on an earlier version of the paper. This work was supported by the National Natural Science Foundation of China (No. 11501061).

References

  1. 1.
    A.O.L. Atkin, Ramanujan congruences for \(p_{-k}(n)\). Can. J. Math. 20, 67–78 (1968)CrossRefGoogle Scholar
  2. 2.
    M.D. Hirschhorn, Partitions in 3 colours. Ramanujan J. 45(2), 399–411 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    M.D. Hirschhorn, The Power of q, Developments in Mathematics, vol. 49 (Springer, Berlin, 2017)Google Scholar
  4. 4.
    N. Saikia, C. Boruah, Congruences of \(\ell \)-regular partition triples for \(\ell \in \{2,3,4,5\}\). Acta Math. Vietnam 42, 551–561 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    C. Wang, S. Chern, Ramanujan-type congruences for 2-color partition triples, arXiv preprint (2017). (arXiv:1706.05667v1)

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingPeople’s Republic of China

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