### Abstract

In 2001, Andrews and Lewis utilized an identity of F. H. Jackson to derive some new partition generating functions as well as identities involving some of the corresponding partition functions. In particular, for 0 < a < b < k, they defined W1(a, b; k; n) to be the number of partitions of n in which the parts are congruent to a or b mod k and such that, for any j, kj + a and kj + b are not both parts. Our primary goal in this note is to prove that W_{1}(1, 3; 4; 27n + 17) ≡ 0 (mod 3) for all n ≥ 0. We prove this result using elementary generating function manipulations and classic results from the theory of partitions.

Original language | English (US) |
---|---|

Article number | 14.9.6 |

Journal | Journal of Integer Sequences |

Volume | 17 |

Issue number | 9 |

State | Published - Sep 4 2014 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Integer Sequences*,

*17*(9), [14.9.6].

}

*Journal of Integer Sequences*, vol. 17, no. 9, 14.9.6.

**A congruence modulo 3 for partitions into distinct non-multiples of four.** / Hirschhorn, Michael D.; Sellers, James Allen.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A congruence modulo 3 for partitions into distinct non-multiples of four

AU - Hirschhorn, Michael D.

AU - Sellers, James Allen

PY - 2014/9/4

Y1 - 2014/9/4

N2 - In 2001, Andrews and Lewis utilized an identity of F. H. Jackson to derive some new partition generating functions as well as identities involving some of the corresponding partition functions. In particular, for 0 < a < b < k, they defined W1(a, b; k; n) to be the number of partitions of n in which the parts are congruent to a or b mod k and such that, for any j, kj + a and kj + b are not both parts. Our primary goal in this note is to prove that W1(1, 3; 4; 27n + 17) ≡ 0 (mod 3) for all n ≥ 0. We prove this result using elementary generating function manipulations and classic results from the theory of partitions.

AB - In 2001, Andrews and Lewis utilized an identity of F. H. Jackson to derive some new partition generating functions as well as identities involving some of the corresponding partition functions. In particular, for 0 < a < b < k, they defined W1(a, b; k; n) to be the number of partitions of n in which the parts are congruent to a or b mod k and such that, for any j, kj + a and kj + b are not both parts. Our primary goal in this note is to prove that W1(1, 3; 4; 27n + 17) ≡ 0 (mod 3) for all n ≥ 0. We prove this result using elementary generating function manipulations and classic results from the theory of partitions.

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UR - http://www.scopus.com/inward/citedby.url?scp=84925682301&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84925682301

VL - 17

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

SN - 1530-7638

IS - 9

M1 - 14.9.6

ER -