Proof of a conjecture involving derangements and roots of unity

Han Wang, Zhi-Wei Sun

Published 2022-06-06Version 1

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.