## Abstract

The notion of broken k-diamond partitions was introduced by Andrews and Paule in 2007. For a fixed positive integer k, let Δ _{k}(n) denote the number of broken k-diamond partitions of n. Recently, Paule and Radu conjectured two relations on Δ _{5}(n) which were proved by Xiong and Jameson, respectively. In this paper, employing these relations, we prove that, for any prime p with p≡1(mod4), there exists an integer λ(p)∈{2,3,5,6,11} such that, for n, α≥ 0 , if p∤ (2 n+ 1) , then (Folmula presented.).Moreover, some non-standard congruences modulo 11 for Δ _{5}(n) are deduced. For example, we prove that, for α≥ 0 , Δ5(11×55α+12)≡7(mod11).

Original language | English (US) |
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Pages (from-to) | 151-159 |

Number of pages | 9 |

Journal | Ramanujan Journal |

Volume | 46 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2018 |

Externally published | Yes |

### Bibliographical note

Funding Information:This work was supported by the National Natural Science Foundation of China (11401260 and 11571143).

Publisher Copyright:

© 2017, Springer Science+Business Media New York.

## Keywords

- Broken k-diamond partition
- Congruence
- Theta function