Minimizing numerical radius of weighted cyclic matrices under permutation of the weights

In this article we answer a question asked by Chien et al. in arXiv:2304.06050 in which they study the numerical range of weighted cyclic matrices under permutation of their entries. Namely, we are interested in how $w(A_\sigma)$ fluctuates for various permutations $\sigma\in S_n$ and fixed $0\leq a_1<\cdots<a_n$ with $A_\sigma=\begin{pmatrix} 0&a_{\sigma(1)}&{}&{}&{}\cr {}&0&a_{\sigma(2)}&{}&{}\cr {}&{}&\ddots&\ddots&{}\cr {}&{}&{}&\ddots&a_{\sigma(n-1)}\cr a_{\sigma(n)}&{}&{}&{}&0 \end{pmatrix}$. Previous results of Gau \cite{gau2024proof} and Chang and Wang \cite{chang2012maximizing} made clear the case when $w(A_\sigma)$ is maximal among all the $w(A_\mu)$ with $\mu\in S_n$. Chien et al. in arXiv:2304.06050 ask what the permutation which makes $w(A_\sigma)$ minimal for $n\geq 6$ could be. Answering this question is the aim of this note.