On the Bessenrodt-Ono inequality for polynomials

Abstract: In 2016, Bessenrodt and Ono proved that the partition function satisfies the inequality $ p(a)p(b)>p(a+b) $ for all $a,b\geqslant2$ with $a+b>9$. Since then, analogous properties have been investigated for many partition statistics. In this talk, following Gian-Carlo Rota's advice, we move from the discrete problem to the continuous one, and consider a family of recursively defined polynomials $ P_n^g(x) := \frac{x}{n} \sum_{k=1}^n g(k) P_{n-k}^g(x) $ with the initial condition $P_0^g(x):=1$, where $(g(n))_{n\in\mathbb{N}}$ is an arbitrary sequence of positive real numbers such that $g(1)=1$. We derive an efficient criterion characterizing when the inequality $ P_{a}^g(x)P_{b}^g(x)\geqslant P_{a+b}^g(x) $ is satisfied for all $x\geqslant x_0$ and $a,b\geqslant1$, where $x_0$ is some real number depending on $g$. Moreover, we illustrate the usefulness of this criterion by applying it to various combinatorial sequences.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)