Some inequalities on ranks of cubic partitions (Q6635547)

This paper explores the combinatorial and analytical properties of cubic partitions, partitions where even parts come into two colors, blue and red. The rank of such partitions, defined as the difference between the count of even parts in blue and red, is analyzed in the context of modular arithmetic. The authors focus on inequalities involving the rank modulo various values of \(m\) and develop generating functions for \(N'(r, m, n)\), the count of cubic partitions of \(n\) whose rank is congruent to \(r\) modulo \(m\). By leveraging \(q\)-series techniques, they establish results for the signs of the differences \(N'(r, m, n) - N'(s, m, n)\) for specific values of \(m\).