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An infinite family of congruences modulo \(3\) for \(13\)-regular bipartitions

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Abstract

Let \(B_{13}(n)\) denote the number of \(13\)-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, we shall prove an infinite family of congruences: for \(\alpha \ge 2\) and \(n\ge 0\),

$$\begin{aligned} B_{13}(3^{\alpha }n+2\cdot 3^{\alpha -1}-1)\equiv \ 0\ (\mathrm{mod\ }3). \end{aligned}$$

In addition, we will also give an alternative proof of one infinite family of congruences for \(b_{13}(n)\), the number of \(13\) regular partitions of \(n\), due to Webb.

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Acknowledgments

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

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Authors and Affiliations

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

    Bernard L. S. Lin

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  1. Bernard L. S. Lin
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Correspondence to Bernard L. S. Lin.

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This work was supported by the Natural Science Foundation of Fujian Province of China (No. 2013J05011) and the Scientific Research Foundation of Jimei University, China.

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Lin, B.L.S. An infinite family of congruences modulo \(3\) for \(13\)-regular bipartitions. Ramanujan J 39, 169–178 (2016). https://doi.org/10.1007/s11139-014-9610-7

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