##
Cyril Banderier,
Christian Krattenthaler,
Alan Krinik, Dmitry Kruchinin, Vladimir Kruchinin, David Tuan Nguyen,
and Michael Wallner

# Explicit formulas for enumeration of lattice paths: basketball and the kernel method

### (41 pages)

**Abstract.**
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*, ..., -1, +1, ..., +*h*.
The case *h*=1 is equivalent to the classical Dyck paths,
for which many ways of getting explicit formulas involving
Catalan-like numbers are known. The case *h*=2 corresponds
to ``basketball'' walks, which we treat in full detail.
Then we move on to the more general case of walks with any finite set
of steps, also allowing some weights/probabilities associated with each step.
We show how a method of wide applicability, the so-called "kernel method",
leads to explicit formulas for the number of walks of length *n*,
for any *h*, in terms of nested sums of binomials.
We finally relate some special cases to other combinatorial problems,
or to problems arising in queuing theory.

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