## 234 Citations

### Variants of the Kernel Method for Lattice Path Models

- Mathematics, Computer Science
- 2014

The iterated kernel method is applied to determine asymptotic information about a family of models in the quarter plane, finding their generating functions explicitly and classifying them as non D-finite.

### Analytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Area †

- Mathematics
- 2006

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…

### Analytic Combinatorics of Lattice Paths : Enumeration and Asymptotics for the Average Area †

- Mathematics
- 2005

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…

### Lattice paths and random walks

- Mathematics
- 2013

Lattice paths are ubiquitous in combinatorics and algorithmics, where they are either studied per se, or as a convenient encoding of other structures. Logically, they play an important role in…

### Asymptotics of lattice walks via analytic combinatorics in several variables

- MathematicsDiscrete Mathematics & Theoretical Computer Science
- 2020

International audience
We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79…

### Lattice paths below a line of rational slope

- MathematicsLattice Path Combinatorics and Applications
- 2019

The approach is extended to the case of generating functions involving several dominant singularities, and has applications to a full class of problems involving some "periodicities", and a key ingredient in the proof is the generalization of an old trick by Knuth himself (for enumerating permutations sortable by a stack).

### Computer Algebra in the Service of Enumerative Combinatorics

- MathematicsISSAC
- 2021

An overview of recent results on structural properties and explicit formulas for generating functions of walks with small steps in the quarter plane are given, especially two important paradigms: "guess-and-prove" and "creative telescoping".

### Classifying lattice walks restricted to the quarter plane

- MathematicsJ. Comb. Theory, Ser. A
- 2009

### Counting lattice paths taking steps in infinitely many directions under special access restrictions

- MathematicsTheor. Comput. Sci.
- 2004

### Generating Functions of Embedded Trees and Lattice Paths

- Computer ScienceFundam. Informaticae
- 2012

The aim of this note is to apply the Bouttier, Di Francesco and Guitter method, consisting of a suitable ansatz and (computer assisted) guessing, to three problems, all related to the enumeration of lattice paths, to derive the generating function of a family of embedded binary trees.

## References

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- MathematicsDiscret. Appl. Math.
- 1999

### The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions

- MathematicsAdvances in Applied Probability
- 2000

Classic works of Karlin and McGregor and Jones and Magnus have established a general correspondence between continuous-time birth-and-death processes and continued fractions of the Stieltjes-Jacobi…

### A Factorization for Formal Laurent Series and Lattice Path Enumeration

- MathematicsJ. Comb. Theory, Ser. A
- 1980

### Random maps, coalescing saddles, singularity analysis, and Airy phenomena

- MathematicsRandom Struct. Algorithms
- 2001

A class of “universal” phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function are exhibited, related to the coalescence of saddle points and the confluence of singularities of generating functions.

### Algebraic geometry for scientists and engineers

- Mathematics
- 1990

Rational and polynomial parametrizations Fractional linear transformations Cubic curves Cubic surfaces and general hypersurfaces Outline of the theory of plane curves Affine plane and projective…

### Reconstruction of lattice sets from their horizontal, vertical and diagonal X-rays

- MathematicsDiscret. Math.
- 2001

### Singularity Analysis of Generating Functions

- MathematicsSIAM J. Discret. Math.
- 1990

This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymptotic…