Sequences with Inequalities

TL;DR

This paper explores inequalities in infinite sequences, particularly focusing on partition numbers, improving existing results, and proving parts of a conjecture related to log-concavity and combinatorial applications.

Contribution

It improves a recent inequality result for partition numbers and proves part of a conjecture related to $ ext{l}$-ary partition numbers.

Findings

Nicolas' log-concavity implies Bessenrodt--Ono inequality for partition numbers.

Several examples illustrating the inequalities are provided.

Partial proof of a conjecture related to $ ext{l}$-ary partition numbers.

Abstract

We consider infinite sequences of positive numbers. The connection between log-concavity and the Bessenrodt--Ono inequality had been in the focus of several papers. This has applications in the white noise distribution theory and combinatorics. We improve a recent result of Benfield and Roy and show that for the sequence of partition numbers $\{p(n)\}$ Nicolas' log-concavity result implies the result of Bessenrodt and Ono towards $p(n) \, p(m) > p(n+m)$. We provide several examples. Benfield and Roy gave a conjecture related to $\ell $-ary partition numbers. We prove part of this conjecture.