Counting lattice paths taking steps in infinitely many directions under special access restrictions

Abstract

We count lattice paths that are confined to the first quadrant by the nature of their step vectors. If no further restrictions apply, a path can go from any point to infinitely many others, but each point on the path has only finitely many predecessors. By “ further restrictions” we mean a boundary line above which the paths may have to stay. Access privilege to the boundary line itself is granted from certain lattice points in the form of a special access step set, which itself may be infinite. We also count the number of paths that contact the weak boundary a given number of times. We approach explicit solutions of such enumeration problems via Sheffer polynomials and functionals, using results of the Umbral Calculus.