Abstract: Given a directed graph in the disc with vertices on the boundary, one can count directed paths from an incoming boundary vertex to an outgoing boundary vertex. If these counts are assembled into a matrix, then Lindstrom's Lemma states that minors of this matrix count certain collections of vertex disjoint paths. In this talk, I will describe some of the remarkable consequences of this simple observation, as well as a long list of generalizations which relate planar combinatorics to various linear algebraic objects. I will conclude with a generalization by Roi Docampo and myself, which parametrizes quasiperiodic linear recurrences by counting certain paths in unoriented planar graphs