A Bessenrodt-Ono inspired inequality for compositions proved constructively

Abstract: In 2016 Bessenrodt-Ono published an analytic proof of the inequality $p(a+b)\leq p(a)p(b)$, where $p(n)$ is the partition function and $a,b$ are positive integers with $a+b>8$. In this talk we consider a similar result for $c(n)$, the number of integer compositions of $n$, and show that $c(a+b)>c(a)c(b)$ for all positive integers $a,b$. Besides numerical verifications, we provide a constructive bijective proof based on the inherent symmetry of compositions. It is known that such a proof is still elusive in the partitions case. We also give an application of our machinery to efficient generation of compositions.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)