I'm trying to calculate the determinant of the following matrix, using the Leibniz formula: $$ \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 & 1 \\ 1 & 0 & 1 & \ddots & \ddots & 0\\ 0 & 1 & \ddots & \ddots & \ddots & \vdots\\ \vdots & \ddots & \ddots & \ddots & 1 & 0\\ 0 & \ddots & \ddots & 1 & 0 & 1\\ 1 & 0 & 0 & 0& 1 & 0 \end{pmatrix}. $$
So, the only permutation that are relevant (non-zero product) are:
$\sigma_1 =
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & ... & n \\
2 & 3 & 4 & 5 & 6 & 7 & ... & 1
\end{pmatrix}$
$\sigma_2 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & ... & n \\ n & 1 & 2 & 3 & 4 & 5 & ... & 1 \end{pmatrix} $
$\sigma_3 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & ... & n-1 & n \\ 2 & 1 & 4 & 3 & 6 & 5 & ... & n & n-1 \end{pmatrix} $
$\sigma_4 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 & n \\ n & 3 & 2 & 5 & 4 & 7 & ... & n-1 & n-2 & 1 \end{pmatrix}$
How would I proceed to determine $\operatorname{Sgn}(\sigma)$ based on the value of $n$?
Thanks!