Strict Log-concavity of $k$-coloured Partitions

TL;DR

This work establishes the strict log-concavity of $p_k(n)$ for all $k\ge 2$, strengthening prior results on partition-analytic inequalities. The authors blend analytic and combinatorial tools, proving a general majorization principle: if a partition ${\bm b}$ majorizes ${\bm a}$, then $p_k({\bm b}) > p_k({\bm a})$ for $k\ge 3$, with a core reduction to length-2 cases via convolutions and Robin Hood transformations. A length-2 case is proved using convolutive properties of $p_k$ and BKRT asymptotics complemented by computer-assisted bounds, which in turn yield corollaries for broader majorization contexts. The paper also outlines open questions on partial majorization and related asymptotic behaviors, providing a framework for future exploration of the structure of $p_k(n)$ under majorization.

Abstract

In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo--Pak, proved that the partition function $p(n)$ is eventually log-concave. Inspired by this and other results, Chern--Fu--Tang first conjectured log-concavity of $k$-coloured partitions. Three of the authors and Tripp later proved this conjecture by introducing recursive sequences and a strict inequality for fractional partition functions, giving explicit errors. In this paper, we show that the log-concavity is, in fact, strict for $k\geq 2$. We shed further light on this phenomenon by utilizing Hardy--Littlewood--Pólya's notion of majorizing. We prove that for partitions $\bm{a},\bm{b}$ of $n\in\N$, if $\bm b$ majorizes $\bm a$, then $p_k(\bm{b})>p_k(\bm{a})$. Numerical calculations indicate that our result is sharp.

Theorems & Definitions (26)

• Theorem 1.1
• Corollary 1.2
• Definition
• Definition
• Theorem 1.3
• Remark
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Remark