Inequalities between overpartition ranks for all moduli

TL;DR

This paper characterizes inequalities between overpartition ranks for all moduli, showing that for large n, certain rank counts are consistently greater, and that sign behaviors depend on residue classes for small moduli.

Contribution

It provides a comprehensive description of inequalities between overpartition ranks across all moduli, extending known results for small moduli to larger ones.

Findings

For c ≥ 7, N̄(a,c,n) > N̄(b,c,n) for large n when 0 ≤ a < b ≤ ⌊c/2⌋.

Sign of rank differences depends on n mod c for small moduli c=2,3,4,5,6.

The behavior for small moduli matches previous findings for c=6, now extended to other small moduli.

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\}. $