Counting Lattice Paths with an Infinite Step Set and Special Access

We count lattice paths that remain in the first quadrant. A path can come from only finitely many lattice points, and if no further restrictions apply, can go to infinitely many others. 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 also may be infinite. Our approach to explicit solutions of such enumeration problems is via Sheffer polynomials and functionals, using results of the Umbral Calculus.