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 $0\le a_1<\cdots <a_n$ with $A_{\sigma }=\left(\begin{array}{ccccc}0& a_{\sigma (1)}& & & \\ & 0& a_{\sigma (2)}& & \\ & & \ddots & \ddots & \\ & & & \ddots & a_{\sigma (n-1)}\\ a_{\sigma (n)}& & & & 0\end{array}\right)$. 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 $n\ge 6$ could be. Answering this question is the aim of this note.