Alan Krinik, Dmitry Kruchinin, Vladimir Kruchinin, David Tuan Nguyen,
and Michael Wallner
Explicit formulas for enumeration of lattice paths: basketball and the kernel method
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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