Polynomization of the Bessenrodt–Ono Type Inequalities for A-Partition Functions

For an arbitrary set or multiset A of positive integers, we associate the A-partition function \(p_A(n)\) $p_A(n)$ (that is the number of partitions of n whose parts belong to A). We also consider the analogue of the k-colored partition function, namely, \(p_{A,-k}(n)\) $p_{A,-k}(n)$ . Further, we define a family of polynomials \(f_{A,n}(x)\) $f_{A,n}(x)$ which satisfy the equality \(f_{A,n}(k)=p_{A,-k}(n)\) $f_{A,n}(k)=p_{A,-k}(n)$ for all \(n\in \mathbb {Z}_{\ge 0}\) $n\in {\mathbb{Z}}_{\ge 0}$ and \(k\in \mathbb {N}\) $k\in \mathbb{N}$ . This paper concerns a polynomialization of the Bessenrodt–Ono inequality, namely \(\begin{aligned} f_{A,a}(x)f_{A,b}(x)>f_{A,a+b}(x), \end{aligned}\) $$\begin{array}{c}\hfill f_{A,a}(x)f_{A,b}(x)>f_{A,a+b}(x),\end{array}$$ where a, b are positive integers. We determine efficient criteria for the solutions of this inequality. Moreover, we also investigate a few basic properties related to both functions \(f_{A,n}(x)\) $f_{A,n}(x)$ and \(f_{A,n}'(x)\) $f_{A,n}^{\prime }(x)$ .