Proof of a conjecture involving derangements and roots of unity

Abstract

Let $n>1$ be an odd integer. For any primitive $n$-th root $\zeta$ of unity in the complex field. Via the Engenvector-eigenvalue Identity, we show that $$\sum_{\tau\in D(n-1)}\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1+\zeta^{j-\tau(j)}}{1-\zeta^{j-\tau(j)}} =(-1)^{\frac{n-1}{2}}\frac{((n-2)!!)^2}{n}, $$ where $D(n-1)$ is the set of all derangements of $1,\ldots,n-1$. This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $\delta=0,1$ we determine the value of $\det[x+m_{jk}]_{1\le j,k\le n}$ completely, where $$m_{jk}=\begin{cases}(1+\zeta^{j-k})/(1-\zeta^{j-k})&\text{if}\ j\not=k,\\\delta&\text{if}\ j=k. \end{cases}$$