### Abstract

In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t¯ (n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd, then it is overlined. In their work, they proved that t¯ (n) satisfies an elegant congruence modulo 3, namely, for n≥ 1 , [Equation not available: see fulltext.]In this work, using elementary tools for manipulating generating functions, we prove that t¯ satisfies a corresponding parity result. We prove that, for all n≥ 1 , t¯(2n)≡{1(mod2)ifn=(3k+1)2for some integerk,0(mod2)otherwise.We also provide a truly elementary proof of the mod 3 characterization of Bringmann et al., as well as a number of additional congruences satisfied by t¯ (n) for various moduli.

Original language | English (US) |
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Journal | Ramanujan Journal |

DOIs | |

State | Published - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Ramanujan Journal*. https://doi.org/10.1007/s11139-019-00156-x

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*Ramanujan Journal*. https://doi.org/10.1007/s11139-019-00156-x

**Congruences for overpartitions with restricted odd differences.** / Hirschhorn, Michael D.; Sellers, James A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Congruences for overpartitions with restricted odd differences

AU - Hirschhorn, Michael D.

AU - Sellers, James A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t¯ (n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd, then it is overlined. In their work, they proved that t¯ (n) satisfies an elegant congruence modulo 3, namely, for n≥ 1 , [Equation not available: see fulltext.]In this work, using elementary tools for manipulating generating functions, we prove that t¯ satisfies a corresponding parity result. We prove that, for all n≥ 1 , t¯(2n)≡{1(mod2)ifn=(3k+1)2for some integerk,0(mod2)otherwise.We also provide a truly elementary proof of the mod 3 characterization of Bringmann et al., as well as a number of additional congruences satisfied by t¯ (n) for various moduli.

AB - In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t¯ (n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd, then it is overlined. In their work, they proved that t¯ (n) satisfies an elegant congruence modulo 3, namely, for n≥ 1 , [Equation not available: see fulltext.]In this work, using elementary tools for manipulating generating functions, we prove that t¯ satisfies a corresponding parity result. We prove that, for all n≥ 1 , t¯(2n)≡{1(mod2)ifn=(3k+1)2for some integerk,0(mod2)otherwise.We also provide a truly elementary proof of the mod 3 characterization of Bringmann et al., as well as a number of additional congruences satisfied by t¯ (n) for various moduli.

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U2 - 10.1007/s11139-019-00156-x

DO - 10.1007/s11139-019-00156-x

M3 - Article

AN - SCOPUS:85069639959

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

ER -