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- W4320932798 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 $$sum_{tauin D(n-1)}mathrm{sign}(tau)prod_{j=1}^{n-1}frac{1+zeta^{j-tau(j)}}{1-zeta^{j-tau(j)}} =(-1)^{frac{n-1}{2}}frac{((n-2)!!)^2}{n}, $$ 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 $det[x+m_{jk}]_{1le j,kle n}$ completely, where $$m_{jk}=begin{cases}(1+zeta^{j-k})/(1-zeta^{j-k})&text{if} jnot=k,delta&text{if} j=k. end{cases}$$" @default.
- W4320932798 created "2023-02-16" @default.
- W4320932798 creator A5030071648 @default.
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- W4320932798 date "2022-06-06" @default.
- W4320932798 modified "2023-09-23" @default.
- W4320932798 title "Proof of a conjecture involving derangements and roots of unity" @default.
- W4320932798 doi "https://doi.org/10.48550/arxiv.2206.02589" @default.
- W4320932798 hasPublicationYear "2022" @default.
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