Explicit Formulas for Enumeration of Lattice Paths: Basketball and the Kernel Method

@article{Banderier2019ExplicitFF,
  title={Explicit Formulas for Enumeration of Lattice Paths: Basketball and the Kernel Method},
  author={Cyril Banderier and Christian Krattenthaler and Alan Krinik and Dmitry V. Kruchinin and Vladimir Kruchinin and David T. Nguyen and Michael Wallner},
  journal={Lattice Path Combinatorics and Applications},
  year={2019}
}
This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude $j$ and ending at a given altitude $k$, with additional constraints such as, for example, to never attain altitude $0$ in-between. We first discuss the case of walks on the integers with steps $-h, \dots, -1, +1, \dots, +h$. The case $h=1$ is equivalent to the classical Dyck paths, for which many ways of getting explicit formulas involving Catalan-like… 
A bijective study of Basketball walks
The Catalan numbers count many classes of combinatorial objects. The most emblematic such objects are probably the Dyck walks and the binary trees, and, whenever another class of combinatorial
Lattice Paths and Submonoids of $$\mathbb Z^2$$
We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set $X\subseteq\mathbb Z^2$, there are two naturally associated
Lattice paths below a line of rational slope
TLDR
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).
Unimodal polynomials and lattice walk enumeration with experimental mathematics
The main theme of this dissertation is retooling methods to work for different situations. I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of
Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for Pushdown Automata
TLDR
A vectorial kernel method is developed which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton for the enumerations of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern.
Lattice Walk Enumeration
Trying to enumerate all of the walks in a 2D lattice is a fun combinatorial problem and there are numerous applications, from polymers to sports. Computers provide a wonderful tool for analyzing
Local time for lattice paths and the associated limit laws
For generalized Dyck paths (i.e., directed lattice paths with any finite set of jumps), we analyse their local time at zero (i.e., the number of times the path is touching or crossing the abscissa).
Bijections between Directed Animals, Multisets and Grand-Dyck Paths
TLDR
This work gives constructive bijections between directed animals, multisets with no consecutive elements and Grand-Dyck paths avoiding the pattern $DUD$, and shows how classical and novel statistics are transported by these bijection.
Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application
TLDR
The main purpose of the research is to derive and improve general methods for developing combinatorial generation algorithms and consider one of these methods, which is based on AND/OR trees.
Euler-Catalan's Number Triangle and Its Application
TLDR
On the basis of properties of Catalan's and Euler's triangles, an explicit formula is obtained that counts the total number of such combinatorial objects and a bivariate generating function.
...
...

References

SHOWING 1-10 OF 73 REFERENCES
Lattice paths below a line of rational slope
TLDR
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).
The Ring of Malcev-Neumann Series and the Residue Theorem
We develop a theory of the field of double Laurent series, iterated Laurent series, and Malcev-Neumann series that applies to most constant term evaluation problems. These include (i) MacMahon's
Random Walks in the Quarter Plane: Algebraic Methods, Boundary Value Problems, Applications to Queueing Systems and Analytic Combinatorics
This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processesarise in numerous
The Number of [Old-Time] Basketball Games with Final Score n:n where the Home Team was Never Losing but also Never Ahead by More Than w Points
TLDR
It is shown that the generating function (in $n$) for the number of walks on the square lattice with steps $(1,1, (1,-1), (2,2)$ and $(2,-2)) satisfies a very special fifth order nonlinear recurrence relation in the region of the lattice.
Formulae and Asymptotics for Coefficients of Algebraic Functions
TLDR
When the function is a power series associated to a context-free grammar, the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers.
Lattice path counting and the theory of queues
Excluded permutation matrices and the Stanley-Wilf conjecture
...
...