An infinite family of congruences modulo $$3$$ 3 for $$13$$ 13 -regular bipartitions

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.