$\det (\mid \mathcal{M}(A_i,E_j)\mid {)}_{i,j}={\sum }_{\pi \in S_r}\mathrm{sgn}(\pi )\cdot \mathrm{\#}\{\text{non-intersecting }\pi \text{-systems}\}$

Expanding by [F1] and reindexing the finite sum by $\pi ={\sigma }^{-1}$ gives $\det (M)={\sum }_{\pi \in S_r}\mathrm{sgn}(\pi ){\prod }_{i<r}{m}_{i,\pi (i)}\mathrm{.}$ Indeed [F3] gives $\mathrm{sgn}({\sigma }^{-1})=\mathrm{sgn}(\sigma )$ and, after putting $j=\sigma (i)$, the product ${\prod }_{i<r}{m}_{\sigma (i),i}$ becomes ${\prod }_{j<r}{m}_{j,{\sigma }^{-1}(j)}$.

For each permutation $\pi$, the product ${\prod }_{i<r}{m}_{i,\pi (i)}$ is exactly the number of $\pi$-systems by [F2], so $\det (M)={\sum }_{\pi \in S_r}\mathrm{sgn}(\pi )\cdot \mathrm{\#}{\mathcal{S}}_{\pi }(A,E)\mathrm{.}$

Split each finite set ${\mathcal{S}}_{\pi }(A,E)$ into its non-intersecting part and its intersecting part. The intersecting systems cancel in pairs under the involution of [L1], because paired terms carry opposite signs and equal absolute values. Therefore the sum of step 2.1 reduces to ${\sum }_{\pi \in S_r}\mathrm{sgn}(\pi )\cdot \mathrm{\#}{\mathcal{N}}_{\pi }(A,E)\mathrm{.}$

In the compatible case, every non-identity permutation $\pi$ has an inversion $i<k$ with $\pi (i)>\pi (k)$, and the compatibility hypothesis says that every path $A_i\to E_{\pi (i)}$ meets every path $A_k\to E_{\pi (k)}$; so every $\pi$-system is intersecting and ${\mathcal{N}}_{\pi }(A,E)=\mathrm{\varnothing }$. Only the identity permutation remains, and the determinant counts the non-intersecting systems joining $A_i$ to $E_i$.