Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats

TL;DR

This paper derives explicit algebraic generating functions and asymptotic formulas for lattice walks in high-dimensional integer grids that avoid backtracking and consecutive steps, with applications to known combinatorial sequences.

Contribution

It provides the first explicit algebraic formulas and polynomial recurrences for complex constrained lattice walks in multiple dimensions.

Findings

Generating functions are algebraic for all considered walk families.

Explicit formulas for walk enumeration are derived.

Connections to Catalan numbers and binomial coefficients are established.

Abstract

We work with lattice walks in $\mathbb{Z}^{r+1}$ using step set $\{\pm 1\}^{r+1}$ that finish with $x_{r+1} = 0$. We further impose conditions of avoiding backtracking (i.e. $[v,-v]$) and avoiding consecutive steps (i.e. $[v,v]$) each possibly combined with restricting to the half-space $x_{r+1} \geq 0$. 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.