I'm reading the following proof of the matrix-tree theorem: https://www.sciencedirect.com/science/article/pii/0012365X9200059Z
in particular, his Theorem 3.1. I was not able to validate the last part: The last lines about equation (3.5)
How do I show the identity $(-1)^{\nu(D)-|W|}=\epsilon(gf)$?
Here $\nu(D)$ stands for the number of connected components of a functional digraph $D$ with $n$ nodes. (A functional digraph is a directed graph whose nodes are all out-degree one.) $|W|$ is, I believe, $t$ of the number of sub-nodes $W=\left\{i_1,\cdots, i_t\right\} \subset \{1,\cdots, n\}$, $L=\left\{j_1,\cdots, j_t\right\} \subset \{1,\cdots, n\}$, and $f\colon W\to L;i_k \mapsto j_k$. The definition of $g$ is slightly involved: Definitions of $W,L,f,g$, and the like
In general, the sign of a permutation of size $n$ is the same as the $(-1)^{n-\text{#cycles}}$ where $\text{#cycles}$ is the number of cycles in the permutation. This can be seen by observing that even-length cycles change the sign while odd-length cycles do not, and that the sum of lengths of cycles is $n$, so the sign of a permutation is determined by the parity of $\sum_C \text{length}(C) - 1 = (n-\text{#cycles})$.
Here $n=|W|$ (by definition) and $\nu(D)=\text{#cycles}(fg)$ (because cycles of the permutation correspond exactly to cycles in the functional graph).