Eryk Lipka, Piotr Miska
The second author studied arithmetic properties of a class of sequences that generalize the sequence of derangements. The aim of the following paper is to disprove two conjectures stated in \cite{miska}. The first conjecture regards the set of prime divisors of their terms. The latter one is devoted to the order of magnitude of considered sequences.
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Let $n>1$ be an odd integer. For any primitive $n$-th root $ζ$ of unity in the complex field. Via the Engenvector-eigenvalue Identity, we show that $$\sum_{τ\in D(n-1)}\mathrm{sign}(τ)\prod_{j=1}^{n-1}\frac{1+ζ^{j-τ(j)}}{1-ζ^{j-τ(j)}} =(-1)^{\frac{n-1}{2}}\frac{((n-2)!!)^2}{n}, $$ 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 $δ=0,1$ we determine the value of $\det[x+m_{jk}]_{1\le j,k\le n}$ completely, where $$m_{jk}=\begin{cases}(1+ζ^{j-k})/(1-ζ^{j-k})&\text{if}\ j\not=k,\\δ&\text{if}\ j=k. \end{cases}$$
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