On 3-regular partitions in 3-colors (Q2327579)

Let \(p_{\{3,k\}}(n)\) denote the number of \(3\)-regular partitions of \(n\) in \(k\)-colors and \(E(q)=\prod_{n\geq 1} (1-q^n)\). In this paper, the authors consider two generating functions in \(q\) for the case of \(p_{3,3}\) evaluated at two special subsequences: \(3^{2\alpha} n+\frac{3^{2\alpha}-1}{4}\) and \(3^{2\alpha+1} n+\frac{3^{2\alpha+2}-1}{4}\). They show that these generating functions can be expressed in terms of \(E(q)\). As a consequence, the authors are able to derive several congruences; as an example they show that for \(\alpha\), \(n\geq 0\) \[ p_{\{3,3\}}\left(3^{2\alpha+1}n+\frac{3^{2\alpha+2}-1}{4}\right)\equiv 0\pmod{3^{2\alpha+2}}. \] Another application is finding the discrete convolution of \(p_{\{3,1\}}(n)\) and \(p_{\{3,2\}}(n)\).