Abstract. This paper develops a unied enumerative and asymptotic theory of directed 2-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are specied by a nite 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 specic \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. To Maurice Nivat, with many thanks for so many things!
|
1446
|
The Art of Computer Programming
– Knuth
- 1973
|
|
649
|
An Introduction to Probability Theory and Its
– Feller
- 1968
|
|
327
|
Combinatorics on Words
– Lothaire
- 1983
|
|
222
|
Singularity analysis of generating functions
– Flajolet, Odlyzko
- 1990
|
|
212
|
Combinatorial Enumeration
– Goulden, Jackson
- 1983
|
|
203
|
An Introduction to the Analysis of Algorithms
– Sedgewick, Flajolet
- 1996
|
|
160
|
Random Walks and Electric Networks
– Doyle, Snell
- 1984
|
|
122
|
Applied and Computational Complex Analysis
– Henrici
- 1974
|
|
117
|
Enumerative Combinatorics, vol. I
– Stanley
- 1997
|
|
105
|
Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable
– Salvy, Zimmermann
- 1994
|
|
95
|
Combinatorial aspects of continued fractions
– Flajolet
- 1980
|
|
87
|
On the altitude of nodes in random trees
– Meir, Moon
- 1978
|
|
78
|
Limit Distributions for Sums of Independent Random Variables (translated from Russian
– Gnedenko, Kolmogorov
- 1954
|
|
71
|
de Bruijn, ``Asymptotic Methods in Analysis
– G
- 1958
|
|
45
|
Analytic Function Theory
– Hille
- 1965
|
|
43
|
On generating functions of generating trees
– Banderier, Bousquet-M'elou, et al.
- 1999
|
|
28
|
The on-line encyclopedia of integer sequences, 2000, Published electronically at http://www.research.att.com/~njas/sequences
– Sloane
|
|
27
|
Algebraic Geometry for Scientists and Engineers
– Abhyankar
- 1990
|
|
22
|
Functional composition patterns and power series reversion
– Raney
- 1960
|
|
20
|
Random maps, coalescing saddles, singularity analysis and Airy
– Banderier, Flajolet, et al.
- 2001
|
|
20
|
La s'erie g'en'eratrice exponentielle dans les probl`emes d"enum'eration. S.M.S
– Foata
- 1974
|
|
20
|
Asymptotic Enumeration Methods, in “Handbook of Combinatorics”, Vol
– Odlyzko
- 1995
|
|
19
|
Reconstructing convex polyominoes from their horizontal and vertical projections, Theor
– Barcucci, Lungo, et al.
- 1996
|
|
18
|
Asymptotic distributions and a multivariate Darboux method in enumeration problems
– Drmota
- 1994
|
|
18
|
Leeuwen (Ed.), Handbook of Theoretical Computer Science, vol. A: Algorithms and Complexity
– van
- 1990
|
|
15
|
A factorization for formal Laurent series and lattice path enumeration
– Gessel
- 1980
|
|
15
|
Lattice path combinatorics with statistical applications
– Narayana
- 1979
|
|
13
|
Sequence of operations analysis for dynamic data structures
– Flajolet, Francon, et al.
- 1980
|
|
13
|
Complex algebraic curves
– Kirwan
- 1992
|
|
11
|
Linear recurrences with constant coecients: the multivariate case
– Bousquet-Melou, Petkovsek
- 2000
|
|
10
|
On the enumeration and generation of generalized Dyck words
– Duchon
- 2000
|
|
8
|
Combinatoire analytique : application aux marches aleatoires
– Banderier
- 1998
|
|
8
|
Verri, Underdiagonal lattice paths with unrestricted steps
– Merlini, Rogers, et al.
- 1999
|
|
6
|
Images and preimages in random mappings
– Drmota, Soria
- 1997
|
|
6
|
Flajolet and Fabrice Guillemin, The formal theory of birth-and-death processes, lattice path combinatorics, and continued fractions
– Philippe
- 2000
|
|
6
|
Brownian motion, bridge, excursion and meander characterized by sampling at independent uniform times
– Pitman
- 1999
|
|
5
|
Observationes analyticae, Novi Commentarii Acad
– Euler
|
|
4
|
Asymptotic properties of some underdiagonal walks generation algorithms, Theoretical Computer Science 218
– Louchard
- 1999
|
|
3
|
Reconstruction of discrete sets from three or more X-rays, Algorithms and complexity
– Barcucci, Brunetti, et al.
- 2000
|
|
3
|
The random generation of directed animals, Theoretical Computer Science 127
– Barcucci, Pinzani, et al.
- 1994
|
|
3
|
On (some) functional equations arising in enumerative combinatorics
– Bousquet-Melou
- 2001
|
|
3
|
Three-dimensional self-avoiding convex polygons
– Bousquet-Melou, Guttmann
- 1997
|
|
3
|
Calcul pratique des coecients de Taylor d'une fonction algebrique, Enseignement Mathematique
– Comtet
- 1964
|
|
3
|
Fayolle and Roudolf Iasnogorodski, Two coupled processors: the reduction to a RiemannHilbert problem, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 47
– Guy
- 1979
|
|
3
|
Roudolf Iasnogorodski, and Vadim Malyshev, Random walks in the quarterplane
– Fayolle
- 1999
|
|
3
|
Algebraic functions over
– Furstenberg
- 1967
|
|
3
|
Theorie analytique des probabilites. Vol
– Laplace
- 1995
|
|
2
|
The number of convex polyominoes reconstructible from their orthogonal projections
– Lungo, Nivat, et al.
- 1994
|
|
2
|
Research bibliography on two number sequences
– Gould
- 1971
|
|
2
|
Sur les series entieres dont la somme est une fonction algebrique, Enseignement mathematique
– Polya
- 1922
|
|
