Inequalities between overpartition ranks for all moduli

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

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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
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

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