it is odd. Geometrically $G$ is a union of disjoint directed cycles. If the order of vertices in all the cycles is fixed, then there are $n!$ ways to assign numbers $\{1 {, \dots ,} n\}$ to the edges; this implies the equality $$\langle W \mid \det_{n,n}^{\varnothing}\rangle = \sum_\sigma
(-1)^{\text{parity of $\sigma$}} w_{1\sigma(1)} \dots w_{n \sigma(n)} =
\det W$$ for any matrix $W = (w_{ij})$. Similarly, for any set $I = \{i_1 {, \dots ,}
i_s\}$ the value $\nolinebreak[0]\langle W \mid \det_{n,n-s}^I\rangle$ is equal to the diagonal minor of the matrix $W$ obtained by deletion of the rows and the columns $i_1 {, \dots ,} i_s$. Also $\langle W \mid
\det_{n,n-1}^{i/j}\rangle$ is equal to the codimension $1$ minor of $W$ obtained by deletion of the row $i$ and the column $j$. This explains the terminology of Definition \[Df:Minors\].

The elements $\det_{n,k}^I$ exhibit some properties one would expect from determinants and minors:

\[Pp:DetProp\]

1.  \[It:RowCol\] (generalized row and column expansion) $$\label{Eq:DetViaMinors}
    \det_{n,k}^{\varnothing}= \frac{1}{k} \sum_{i,j=1}^n ([ij]) *
    \det_{n,k-1}^{i/j}.$$

2.  \[It:PDer\] (partial derivative with respect to a diagonal matrix element) Let matrix elements $w_{ij}$, $i,j = 1 {, \dots ,} n$, of the matrix $W$ be independent (commuting) variables. Then for any $i 