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

摘要

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(An + B)) and Δ3(8(An + B) + 6) is infinitely often even and infinitely often odd. In this sense, we generalize Subbaraoʼs Conjecture for this function Δ3. To the best of our knowledge, this is the first generalization of Subbaraoʼs Conjecture in the literature.