An extensive analysis of the parity of broken 3-diamond partitions

In 2007, Andrews and Paule introduced the family of functions [Formula: see text] which enumerate the number of broken k-diamond partitions for a fixed positive integer k. Since then, numerous mathematicians have considered partitions congruences satisfied by [Formula: see text] for small values of k. In this work, we provide an extensive analysis of the parity of the function [Formula: see text], including a number of Ramanujan-like congruences modulo 2. This will be accomplished by completely characterizing the values of [Formula: see text] modulo 2 for [Formula: see text] and any value of [Formula: see text]. In contrast, we conjecture that, for any integers [Formula: see text], [Formula: see text] and [Formula: see text] is infinitely often even and infinitely often odd. In this sense, we generalize Subbarao's Conjecture for this function [Formula: see text]. To the best of our knowledge, this is the first generalization of Subbarao's Conjecture in the literature.