## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 3703-3716 |

Number of pages | 14 |

Journal | Journal of Number Theory |

Volume | 133 |

Issue number | 11 |

DOIs | |

State | Published - 2013 |

Externally published | Yes |

## Keywords

- Broken k-diamonds
- Congruences
- Modular forms
- Partitions