We consider some q-series which depend on a pair of positive integers (k, m). While positivity of these series holds for the first few values of (k, m), the situation is quite unclear for other values of (k, m). In addition, our series generate the number of certain two-color integer partitions weighted by (-1) ʲ where j is the number of even parts. Therefore, inequalities involving these partitions will be deduced from the positivity of their generating functions.
Andrews et al. (Sat,) studied this question.