## Abstract

In 2007, Andrews and Paule introduced the family of functions Δ_{k}(n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, Δ_{1}(2n+1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized this combinatorial approach to naturally define a generalization of broken k-diamond partitions which he called k dots bracelet partitions. He denoted the number of k dots bracelet partitions of n by _{k}(n) and proved various congruence properties for these functions modulo primes and modulo powers of 2. In this note, we extend the set of congruences proven by Fu by proving the following congruences: For all n ≥ 0, $$\begin{array}{r@{}cl}\mathfrak{B}-5(10n+7) &\equiv& 0 \pmod{5 2},\\[4pt]\mathfrak{B}-7(14n+11) &\equiv& 0 \pmod{7 2}, \quad {\rm and}\\[4pt]\mathfrak{B}-{11}(22n+21) &\equiv& 0 \pmod{11 2}\end{array}$$ We also conjecture an infinite family of congruences modulo powers of 7 which are satisfied by the function _{7}.

Original language | English (US) |
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Pages (from-to) | 939-943 |

Number of pages | 5 |

Journal | International Journal of Number Theory |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2013 |

Externally published | Yes |

### Bibliographical note

Funding Information:C.-S. Radu was funded by the Austrian Science Fund (FWF), W1214-N15, project DK6 and by grant P2016-N18. The research was supported by the strategic program “Innovatives OÖ 2010 plus” by the Upper Austrian Government.

## Keywords

- Broken k-diamonds
- congruences
- k dots bracelet partitions
- modular forms
- partitions