## Basic Analytic Combinatorics of Directed Lattice Paths (2001)

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Venue: | Theoretical Computer Science |

Citations: | 58 - 10 self |

### BibTeX

@ARTICLE{Banderier01basicanalytic,

author = {Cyril Banderier and Philippe Flajolet},

title = {Basic Analytic Combinatorics of Directed Lattice Paths},

journal = {Theoretical Computer Science},

year = {2001},

volume = {281},

pages = {1--2}

}

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### Abstract

This paper develops a unified enumerative and asymptotic theory of directed 2-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially 1-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.