A further look at cubic partitions

The partitions in which even parts come in two colours are known as cubic partitions. In this paper, we introduce and investigate the cubic partition function A(n) which is defined as the difference between the number of cubic partitions of n into an even numbers of parts and the number of cubic partitions of n into an odd numbers of parts. We present a collection of identities relating A(n) and provide analytic proofs. Among these identities we remark two Ramanujan-like congruences: for all $$n\geqslant 0$$ , $$A(9n+5) \equiv 0 \pmod 3$$ and $$A(27n+26) \equiv 0 \pmod 3$$ . Combinatorial interpretations for A(n) are given in terms of the ordinary partitions into parts congruent to $$\pm 1, 4 \pmod 8$$ when n is even and congruent to $$\pm 3, 4 \pmod 8$$ when n is odd. In this context, some infinite families of linear inequalities involving A(n) are proposed as open problems.