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

In 2007, Andrews and Paule introduced the family of functions _δk(n) 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 _δk(n) for small values of k. In this work, we provide an extensive analysis of the parity of the function _δ3(n), including a number of Ramanujan-like congruences modulo 2. This will be accomplished by completely characterizing the values of _δ3(8n + r) modulo 2 for r ∈ {1, 2, 3, 4, 5, 7} and any value of n ≥ 0. In contrast, we conjecture that, for any integers 0 ≤ B < A, _δ3(8(A n + B)) and _δ3(8(A n + B) + 6) is infinitely often even and infinitely often odd. In this sense, we generalize Subbarao's Conjecture for this function _δ3. To the best of our knowledge, this is the first generalization of Subbarao's Conjecture in the literature.