= Nonintersecting lattice-path determinant
For a weighted acyclic directed graph, let $M_{ij}$ be the sum of path weights from $A_j$ to $B_i$. Expanding $\det M$ gives a signed sum of path families. At the first intersection in a fixed topological order, swap the two path tails. The operation preserves weight and reverses the endpoint <permutation>'s sign, so intersecting families cancel. Only vertex-disjoint families remain. If planarity forces these families to join identically indexed endpoints, their sum has positive sign. Applied to northeast paths with horizontal weight $x_r$ at height $r$, this gives the skew <Jacobi–Trudi identity>.
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