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Strict log-concavity of k-coloured partitions
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 18 May 2026, pp. 1-15
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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 the 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 
$\boldsymbol{a},\boldsymbol{b}$ of 
$n\in{\mathbb N}$, if 
$\boldsymbol b$ majorizes 
$\boldsymbol a$, then 
$p_k(\boldsymbol{a}) \gt p_k(\boldsymbol{b})$. Numerical calculations indicate that our result is sharp.
