# Congruences modulo powers of 3 for 2-color partition triples

Article

First Online:

- 7 Downloads

## Abstract

Let \(p_{k,3}(n)\) enumerate the number of 2-color partition triples of and

*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$$\begin{aligned} p_{3,3}\left( 3^{\alpha }n+\dfrac{3^{\alpha }+1}{2}\right)&\equiv 0\pmod {3^{\alpha +1}} \end{aligned}$$

$$\begin{aligned} p_{3,3}\left( 3^{\alpha +1}n+\dfrac{5\times 3^{\alpha }+1}{2}\right)&\equiv 0\pmod {3^{\alpha +4}}. \end{aligned}$$

## 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.A.O.L. Atkin, Ramanujan congruences for \(p_{-k}(n)\). Can. J. Math.
**20**, 67–78 (1968)CrossRefGoogle Scholar - 2.M.D. Hirschhorn, Partitions in 3 colours. Ramanujan J.
**45**(2), 399–411 (2018)MathSciNetCrossRefGoogle Scholar - 3.M.D. Hirschhorn,
*The Power of q, Developments in Mathematics*, vol. 49 (Springer, Berlin, 2017)Google Scholar - 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.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