$M_2$-Ranks of overpartitions modulo $6$ and $10$

TL;DR

This paper investigates inequalities related to the $M_2$-ranks of overpartitions modulo 6 and 10, deriving generating functions and expressing some in terms of mock theta functions, thus advancing understanding of overpartition rank distributions.

Contribution

The paper derives new inequalities for $M_2$-ranks of overpartitions modulo 6 and 10 and expresses related generating functions using mock theta functions, extending previous work on rank inequalities.

Findings

Established inequalities for overpartition $M_2$-ranks modulo 6.

Expressed certain generating functions in terms of mock theta functions.

Derived formulas for rank difference generating functions using modular function methods.

Abstract

In this paper, we obtain inequalities on $M_2$-ranks of overpartitions modulo $6$. Let $\overline{N}_2(s,m,n)$ to be the number of overpartitions of $n$ whose $M_2$-rank is congruent to $s$ modulo $m$. For $M_2$-ranks modulo $3$, Lovejoy and Osburn derived the generating function of $\overline{N}_2(s,3,n)-\overline{N}_2(t,3,n)$, which implies the inequalities $\overline{N}_2(0,3,n)\geq\overline{N}_2(1,3,n)$. For $\ell=6, 10$, we consider the generating function $\overline{R}_{s,t}(d,\ell)$ of the $M_2$-rank differences $\overline{N}_2(s,\ell,\ell n/2+d) + \overline{N}_2(s+1,\ell,\ell n/2+d) - \overline{N}_2(t,\ell,\ell n/2+d) - \overline{N}_2(t+1,\ell,\ell n/2+d)$. By the method of Lovejoy and Osburn, we derive a formula for $\overline{R}_{0,2}(d,6)$. This leads to the inequalities for $n\geq0$, $\overline{N}_2(0,6,3n)\geq\overline{N}_2(2,6,3n)$ and $\overline{N}_2(0,6,3n+1) \geq \overline{N}_2(2,6,3n+1)$. Based on the valence formula for modular functions, we compute $\overline{R}_{0,4}(d,10)$ and $\overline{R}_{1,3}(d,10)$. In particular, we notice that the generating function $\overline{R}_{0,2}(2,6)$ can be expressed in terms of the third order mock theta function $\rho(q)$, and the generating functions $\overline{R}_{0,4}(4,10)$, $\overline{R}_{1,3}(1,10)$ and $\overline{R}_{1,3}(4,10)$ can also be expressed in terms of the tenth order mock theta functions $\phi(q)$ and $\psi(q)$.