English

Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats

Combinatorics 2021-11-11 v2

Abstract

We work with lattice walks in Zr+1\mathbb{Z}^{r+1} using step set {±1}r+1\{\pm 1\}^{r+1} that finish with xr+1=0x_{r+1} = 0. We further impose conditions of avoiding backtracking (i.e. [v,v][v,-v]) and avoiding consecutive steps (i.e. [v,v][v,v]) each possibly combined with restricting to the half-space xr+10x_{r+1} \geq 0. We find in all cases the generating functions for such walks are algebraic and give explicit formulas for them. We also find polynomial recurrences for their coefficients. From the generating functions we find the asymptotic enumeration of each family of walks considered. The enumeration in special cases includes central binomial coefficients and Catalan numbers as well as relations to enumeration of another family of walks previously studied for which we provide bijection.

Keywords

Cite

@article{arxiv.2105.02417,
  title  = {Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats},
  author = {John Machacek},
  journal= {arXiv preprint arXiv:2105.02417},
  year   = {2021}
}
R2 v1 2026-06-24T01:49:29.764Z