Generating functions and duality for non-crossing walks on a plane graph.

The generating function for the number of non-crossingwalk configurations of n walks between the roots of a two-rooted directed plane graph is introduced. This is shown to be a rational function and the structure and symmetry property of its numerator are discussed. The walk configurations correspond to flows and the equivalent dual generating function for potentials is investigated independently. Also equivalences are drawn with the partially order sets that can be constructed from the walk configurations. Finally, the general results developed here are applied to the directed square and honeycomb lattices. [ABSTRACT FROM AUTHOR]