Log-concavity of the restricted partition function $p_\mathcal{A}(n,k)$ and the new Bessenrodt-Ono type inequality

TL;DR

This paper establishes a new inequality for the restricted partition function and characterizes conditions for its log-concavity, using asymptotic analysis and classical estimation techniques.

Contribution

It introduces a novel Bessenrodt-Ono type inequality for $p_ ext{A}(n,k)$ and determines when the sequence is log-concave based on asymptotic behavior.

Findings

New Bessenrodt-Ono type inequality for $p_ ext{A}(n,k)$

Conditions for log-concavity of $p_ ext{A}(n,k)$ sequences

Application of asymptotic analysis and classical estimations

Abstract

Let $\mathcal{A}=(a_i)_{i=1}^\infty$ be a non-decreasing sequence of positive integers and let $k\in\mathbb{N}_+$ be fixed. The function $p_\mathcal{A}(n,k)$ counts the number of partitions of $n$ with parts in the multiset $\{a_1,a_2,\ldots,a_k\}$. We find out a new type of Bessenrodt-Ono inequality for the function $p_\mathcal{A}(n,k)$. Further, we discover when and under what conditions on $k$, $\{a_1,a_2,\ldots,a_k\}$ and $N\in\mathbb{N}_+$, the sequence $\left(p_\mathcal{A}(n,k)\right)_{n=N}^\infty$ is log-concave. Our proofs are based on the asymptotic behavior of $p_\mathcal{A}(n,k)$, in particular, we apply the results of Netto and P\'olya-Szeg\"o as well as the Almkavist's estimation.