Let $n>1$ be an odd integer, and let $\zeta$ be a primitive $n$th root of unity in the complex field. Via the Eigenvector-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 $\det[x+m_{jk}]_{1\leqslant j,k\leqslant n-1}$ completely, where$$m_{jk}=\begin{cases}(1+\zeta^{j-k})/(1-\zeta^{j-k})&\text{if}\ j\not=k,\\\delta&\text{if}\ j=k.\end{cases}$$
@article{10_37236_11377,
author = {Han Wang and Zhi-Wei Sun},
title = {Proof of a conjecture involving derangements and roots of unity},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {2},
doi = {10.37236/11377},
zbl = {1508.05013},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11377/}
}
TY - JOUR
AU - Han Wang
AU - Zhi-Wei Sun
TI - Proof of a conjecture involving derangements and roots of unity
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/11377/
DO - 10.37236/11377
ID - 10_37236_11377
ER -
%0 Journal Article
%A Han Wang
%A Zhi-Wei Sun
%T Proof of a conjecture involving derangements and roots of unity
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/11377/
%R 10.37236/11377
%F 10_37236_11377
Han Wang; Zhi-Wei Sun. Proof of a conjecture involving derangements and roots of unity. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/11377
