Using Grenet's construction (http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=193FC514DC541C17E8F03A7A6BDE4C61?doi=10.1.1.717.4014&rep=rep1&type=pdf ) we can write the 3x3 permanent as the 7x7 determinant as
my question is, how does one determine how the labels for the edge weighting work? This seems to be explained in Peter Burgisser's article (https://www.mat.univie.ac.at/~slc/wpapers/s75buerg.pdf) on page 3 sort of seems to describe how to assign the edge labels but I dont follow the arguement entirely. I am confused on what the algorithm is for the edge labelling in the general case of a nxn permanent corresponding to an adjacency matrix of a hypercube
The labels on the edges work as follows. We want to compute the permanent of an n x n- matrix with the entries $x_{ij}$. The diagram (Hasse diagram of the power set of $\{1,...,n\}$, ordered by reverse inclusion) has layers. Layer 0 consists of the empty set, layer k of k-element sets.
Suppose we have an edge that goes from the node A to the node B. A and B are subsets of $\{1,...,n\}$ and B has exactly one element more than A. Let's say $j$ is this new element and B has exactly $i$ elements, i.e., B is on layer $i$. In this case the edge is labeled by $x_{ij}$.
In order to compute the permanent of the matrix $(x_{ij})$ you take all the paths from $\emptyset$ to $\{1,...,n\}$ and multiply the labels along each path. Note that each path chooses exactly one entry of the matrix in each row. Now you add up these products for all paths. Voila, there is your permanent.