Strict Log-concavity of $k$-coloured Partitions

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 $k\ge 2$. We shed further light on this phenomenon by utilizing Hardy--Littlewood--Pólya's notion of majorizing. We prove that for partitions $\mathbf{a},\mathbf{b}$ of $n\in \mathbb{N}$, if $\mathbf{b}$ majorizes $\mathbf{a}$, then $p_k(\mathbf{b})>p_k(\mathbf{a})$. Numerical calculations indicate that our result is sharp.