Basic analytic combinatorics of directed lattice paths
نویسندگان
چکیده
This paper develops a uni(ed enumerative and asymptotic theory of directed two-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are speci(ed by a (nite 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 speci(c “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. c © 2002 Elsevier Science B.V. All rights reserved.
منابع مشابه
Some reflections on directed lattice paths
This article analyzes directed lattice paths, when a boundary reflecting or absorbing condition is added to the classical models. The lattice paths are characterized by two time-independent sets of rules (also called steps) which have a privileged direction of increase and are therefore essentially one-dimensional objects. Depending on the spatial coordinate, one of the two sets of rules applie...
متن کاملAnalytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Average Area†
This paper tackles the enumeration and asymptotics of the area below directed lattice paths (walks on N, with a finite set of jumps). It is a nice surprise (obtained via the “kernel method”) that the generating functions of the moments of the area are algebraic functions, expressible as symmetric functions in terms of the roots of the kernel. For a large class of walks, we give full asymptotics...
متن کاملAnalytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Area
This paper tackles the enumeration and asymptotics of the area below directed lattice paths (walks on N, with a finite set of jumps). It is a nice surprise (obtained via the “kernel method”) that the generating functions of the moments of the area are algebraic functions, expressible as symmetric functions in terms of the roots of the kernel. For a large class of walks, we give full asymptotics...
متن کاملOn independent domination numbers of grid and toroidal grid directed graphs
A subset $S$ of vertex set $V(D)$ is an {em indpendent dominating set} of $D$ if $S$ is both an independent and a dominating set of $D$. The {em indpendent domination number}, $i(D)$ is the cardinality of the smallest independent dominating set of $D$. In this paper we calculate the independent domination number of the { em cartesian product} of two {em directed paths} $P_m$ and $P_n$ for arbi...
متن کاملCombinatorics of lattice paths with and without spikes
We derive a series of results on random walks on a d-dimensional hypercubic lattice (lattice paths). We introduce the notions of terse and simple paths corresponding to the path having no backtracking parts (spikes). These paths label equivalence classes which allow a rearrangement of the sum over paths. The basic combinatorial quantities of this construction are given. These formulas are usefu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Theor. Comput. Sci.
دوره 281 شماره
صفحات -
تاریخ انتشار 2002