32 Lattice paths and determinants
Let \(G = (V, E)\) be a finite weighted acyclic directed graph, \(A = \{ A_1, \dots , A_n\} \) and \(\mathcal{B} = \{ B_1, \dots , B_n\} \) two \(n\)-sets of vertices, and \(M\) the path matrix from \(A\) to \(\mathcal{B}\). Then
\[ \det M = \sum _{\mathcal{P} \text{ vertex-disjoint path system}} \text{sign}(\mathcal{P}) \, w(\mathcal{P}). \tag {3} \]
Proof
TODO
Let \(G = (V, E)\) be a finite weighted acyclic directed graph, \(A = \{ A_1, \dots , A_n\} \) and \(\mathcal{B} = \{ B_1, \dots , B_n\} \) two \(n\)-sets of vertices, and \(M\) the path matrix from \(A\) to \(\mathcal{B}\). Then
\[ \det M = \sum _{\mathcal{P} \text{ vertex-disjoint path system}} \text{sign}(\mathcal{P}) \, w(\mathcal{P}). \tag {3} \]
Proof
TODO