Inequalities for $k$-regular partitions

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 $E_{\mathrm{\infty }}$ and $F_{\mathrm{\infty }}$ for the partition function $p(n)$. We prove by induction that $E_k=E_{\mathrm{\infty }}$ and $F_k=F_{\mathrm{\infty }}$ if and only if $k\ge 10$. Beckwith and Bessenrodt used analytic methods to consider $2\le k\le 6$, 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.