Congruences modulo 27 for cubic partition pairs (Q331097)

Let \(a(n)\) denote the number of cubic partitions of \(n\), i.e., the number of partitions of \(n\) such that the even parts can appear in two colours. Its generating function is NEWLINE\[NEWLINE\prod^\infty_{j=1} {1\over(1-q^j)(1-q^{2j})}.NEWLINE\]NEWLINE Let \(b(n)\) denote the number of cubic partition pairs of \(n\) in the sense that its generating function is NEWLINE\[NEWLINE\prod^\infty_{j=1} {1\over(1-q^j)^2(1-q^{2j})^2}.NEWLINE\]NEWLINE \textit{H. Zhao} and \textit{Z. Zhong} [Electron. J. Comb. 18, No. 1, Research Paper P58, 9 p. (2011; Zbl 1220.05006)] established several Ramanujan type congruences modulo 5, 7, and 9 for \(b(n)\), e.g., \(b(9n+7)\equiv 0\pmod 9\).NEWLINENEWLINE In the paper under review, the author proves that \(b(27n+16)\equiv 0\pmod{27}\), \(b(27n+25)\equiv 0\pmod{27}\), \(b(81n+61)\equiv 0\pmod{27}\). Then many infinite families of congruences modulo 27 for \(b(n)\) are presented. The author also proposes conjectures on congruences for \(b(n)\) modulo 49, 81, and 243.