The number of sequences of odd length in strict partitions (denoted as \(\textrm{sol}\) ), which plays a pivotal role in the first paper of this series, is investigated in different contexts, both new and old. Namely, we first note a direct link between \(\textrm{sol}\) and the 2-measure of strict partitions when the partition length is given. This notion of 2-measure of a partition was introduced quite recently by Andrews, Bhattacharjee, and Dastidar. We establish a q-series identity in three ways: one of them features a Franklin-type involution. Secondly, still with this new partition statistic \(\textrm{sol}\) in mind, we revisit Euler’s partition theorem through the lens of Sylvester–Bessenrodt. Two new bivariate refinements of Euler’s theorem are established, which involve notions such as MacMahon’s 2-modular Ferrers diagram, the Durfee side of partitions, and a certain alternating index of partitions that we believe is introduced here for the first time.
