### 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) |
---|---|

Pages (from-to) | 151-159 |

Number of pages | 9 |

Journal | Ramanujan Journal |

Volume | 46 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Ramanujan Journal*,

*46*(1), 151-159. https://doi.org/10.1007/s11139-017-9894-5

}

*Ramanujan Journal*, vol. 46, no. 1, pp. 151-159. https://doi.org/10.1007/s11139-017-9894-5

**Congruences modulo 11 for broken 5-diamond partitions.** / Liu, Eric H.; Sellers, James Allen; Xia, Ernest X.W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Congruences modulo 11 for broken 5-diamond partitions

AU - Liu, Eric H.

AU - Sellers, James Allen

AU - Xia, Ernest X.W.

PY - 2018/5/1

Y1 - 2018/5/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=85018290079&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018290079&partnerID=8YFLogxK

U2 - 10.1007/s11139-017-9894-5

DO - 10.1007/s11139-017-9894-5

M3 - Article

AN - SCOPUS:85018290079

VL - 46

SP - 151

EP - 159

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

IS - 1

ER -