Enumeration of lattice paths using finite operator calculus

This dissertation discusses umbral calculus and lattice path enumeration and then continues by explicitly enumerating weighted directed lattice paths staying above a boundary using finite operator calculus. In Part I we discuss the history and representative results of the two topics. We separate umbral calculus into two fields, classical umbral calculus and finite operator calculus, and attempt to correct their intertwined histories. We discuss the beginnings of lattice path enumeration and survey the types of lattice path enumeration problems and solution methods found in the literature. In Part II, we give necessary conditions of a step set or of its equivalent operator equation such that the path count functions coincide with Sheffer polynomials where the path counts are nonzero. We derive the polynomials from an expansion theorem that includes a polynomial basis and initial conditions. The polynomial basis is derived from a known basic sequence with a transfer formula and a linear operator equation based on the step set. The initial conditions are functionals on the polynomials designed to vanish when evaluated along the boundary line for all but finitely many values. We solve lattice path enumeration problems with four types of boundary conditions and various step sets. We work out general solutions for paths that stay in the first quadrant, paths that stay in the first quadrant and above a line with an integer slope, and paths that can reach the boundary with an additional privileged access step set. We count the number of paths, and in one example we count the paths refined by the number of times they contact the boundary. We explore step sets including a general three-element step set, weighted finite step sets, weighted infinite step sets, and step sets that include paths as steps called pathlets. We research if our methods still give explicit solutions as we complicate and expand the step sets. The example sections include fourteen explicitly worked out problems. Part II of the dissertation includes and extends the three papers on the subject by Humphreys and Niederhausen written between 2000 and 2004.