Polynomization of the Bessenrodt-Ono inequality

In this paper we investigate the generalization of the Bessenrodt--Ono inequality by following Gian-Carlo Rota's advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of $k$-colored partitions of $n$ as special values of polynomials $P_n(x)$. We prove for all real numbers $x >2$ and $a,b \in \mathbb{N}$ with $a+b >2$ the inequality \begin{equation*} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{equation*} We show that $P_n(x) < P_{n+1}(x)$ for $x \geq 1$, which generalizes $p(n) < p(n+1)$, where $p(n)$ denotes the partition function. Finally, we observe for small values, the opposite can be true since for example: $P_2(-3+ \sqrt{10}) = P_{3}(-3 + \sqrt{10})$.