Ciolan, Alexandru
(2021)
*Inequalities between overpartition ranks for all moduli.*
MPIM Preprint Series 2021 (4).

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## Abstract

In this paper we give a full description of the inequalities that can occur between overpartition ranks. If $ \overline N(a,c,n) $ denotes the number of overpartitions of $ n $ with rank congruent to $ a $ modulo $ c,$ we prove that for any $ c\ge7 $ and $ 0\le a<b\le\left\lfloor\frac{c}{2}\right\rfloor $ we have $ \overline N(a,c,n)>\overline N(b,c,n) $ for $n$ large enough. That the sign of the rank differences $ \overline N(a,c,n)-\overline N(b,c,n) $ depends on the residue class of $ n $ modulo $ c $ in the case of small moduli, such as $ c=6, $ is known due to the work of Ji, Zhang and Zhao (2018) and Ciolan (2020). We show that the same behavior holds for $c\in\{2,3, 4,5\}$.

Item Type: | MPIM Preprint |
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Subjects: | 1 Discrete mathematics / algebra > 11-XX Number theory |

Divisions: | Research > Preprints |

Depositing User: | Andrea Kohlhuber |

Date Deposited: | 11 Feb 2021 11:29 |

Last Modified: | 15 Feb 2021 12:11 |

URI: | https://archive.mpim-bonn.mpg.de/id/eprint/4197 |

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