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Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats

Authors :
John Machacek
Source :
Enumerative Combinatorics and Applications, Vol 2, Iss 1, p Article S2R3 (2021)
Publication Year :
2021
Publisher :
University of Haifa, 2021.

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.

Details

ISSN :
27102335
Volume :
2
Database :
OpenAIRE
Journal :
Enumerative Combinatorics and Applications
Accession number :
edsair.doi.dedup.....1f6b5a87da3c7bd216b2f22d758aee56