On the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions

The $A$-partition function $p_A(n)$ enumerates those partitions of $n$ whose parts belong to a fixed (finite or infinite) set $A$ of positive integers. On the other hand, the extended $A$-partition function $p_A\left(\boldsymbol{\mu}\right)$ is defined as an multiplicative extension of the $A$-partition function to a function on $A$-partitions. In this paper, we investigate the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions. In particular, we examine the property for both the $m$-ary partition function $b_m(n)$ and the $d$-th power partition function $p_d(n)$. Moreover, we show that $b_m(\boldsymbol{\mu})$ ($p_d(\boldsymbol{\mu})$) takes its maximum value at an explicitly described set of $m$-ary partitions (power partitions), where $\boldsymbol{\mu}$ is an $m$-ary partition (a power partition) of $n$. Additionally, we exhibit analogous results for the Fibonacci partition function and the `factorial' partition function. It is worth pointing out that an elementary combinatorial reasoning plays a crucial role in our investigation.