Basic analytic combinatorics of directed lattice paths

Abstract

This paper develops a unified enumerative and asymptotic theory of directed two-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are specified by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially one-dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths. © 2002 Elsevier Science B.V. All rights reserved.