• DOI:10.1007/978-3-030-11102-1_6
• Corpus ID: 119142474

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}
}

• C. Banderier, C. Krattenthaler, +4 authors M. Wallner
• Published 21 September 2016
• Mathematics
• Lattice Path Combinatorics and Applications

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 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.