Recently, Andrews and Hopkins provided a partial extension of the Crank-Mex theorem by refining the equinumerosity among partitions with even mex, with a fixed point, with negative crank and with positive crank in terms of the number of parts greater than one. In this work, we first give a further refinement of Andrews and Hopkins' result by including the 2-measure and the size of the Durfee square or rectangle, and then establish a complementary “odd-mex” counterpart.
Wang et al. (2026) studied this question.