Inequalities for $k$-regular partitions

TL;DR

This paper investigates inequalities related to $k$-regular partition functions, identifying when certain inequalities hold or are equal, and extends conjectures on their log-concavity, building on prior work for specific $k$ values.

Contribution

It determines the sets of solutions for the Bessenrodt--Ono inequality for all $k$, proving their equivalence to known cases for $k \\geq 10$, and proposes a comprehensive conjecture on log-concavity.

Findings

Sets $E_k$ and $F_k$ match known cases for $k \\geq 10$

Established conditions for equality and inequality in $k$-regular partitions

Proposed a detailed conjecture on the log-concavity of $p_k(n)$

Abstract

We build upon the work by Bessenrodt and Ono, as well as Beckwith and Bessenrodt concerning the combined additive and multiplicative behavior of the $k$-regular partition functions $p_k(n)$. Our focus is on addressing the solutions of the Bessenrodt--Ono inequality \begin{equation*} p_k(a) \, p_k(b) > p_k(a+b). \end{equation*} We determine the sets $E_k$ and $F_k$ consisting of all pairs $(a,b)$, where we have equality or the opposite inequality. Bessenrodt and Ono previously determined the exception sets $E_{\infty}$ and $F_{\infty}$ for the partition function $p(n)$. We prove by induction that $E_k=E_{\infty}$ and $F_k=F_{\infty}$ if and only if $k \geq 10$. Beckwith and Bessenrodt used analytic methods to consider $2 \leq k \leq 6$, while Alanazi, Gagola, and Munagi studied the case $k=2$ using combinatorial methods. Finally, we present a precise and comprehensive conjecture on the log-concavity of the $k$-regular partition function extending previous speculations by Craig and Pun. The case $k=2$ was recently proven by Dong and Ji.