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

Pages (from-to) | 3703-3716 |

Number of pages | 14 |

Journal | Journal of Number Theory |

Volume | 133 |

Issue number | 11 |

DOIs | |

State | Published - Jul 30 2013 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*133*(11), 3703-3716. https://doi.org/10.1016/j.jnt.2013.05.009

}

*Journal of Number Theory*, vol. 133, no. 11, pp. 3703-3716. https://doi.org/10.1016/j.jnt.2013.05.009

**An extensive analysis of the parity of broken 3-diamond partitions.** / Radu, Silviu; Sellers, James A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An extensive analysis of the parity of broken 3-diamond partitions

AU - Radu, Silviu

AU - Sellers, James A.

PY - 2013/7/30

Y1 - 2013/7/30

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84880618645&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880618645&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2013.05.009

DO - 10.1016/j.jnt.2013.05.009

M3 - Article

AN - SCOPUS:84880618645

VL - 133

SP - 3703

EP - 3716

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 11

ER -