The vectorial kernel method for walks with longer steps

TL;DR

This paper extends the vectorial kernel method to analyze lattice paths with longer steps, providing new applications and proving a conjecture on the asymptotic behavior of ascents in Schroeder paths.

Contribution

The paper generalizes the vectorial kernel method to walks with longer steps, expanding its applicability to more complex path constraints.

Findings

Extended the vectorial kernel method to longer steps in lattice paths.

Applied the extended method to new combinatorial problems.

Proved a conjecture about the asymptotic behavior of ascents in Schroeder paths.

Abstract

Asinowski, Bacher, Banderier and Gittenberger (A. Asinowski, A. Bacher, C. Banderier and B. Gittenberger. Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata. Algorithmica, pp. 1-43, 2019.) recently developed the vectorial kernel method - a powerful extension of the classical kernel method that can be used for paths that obey constraints that can be described by finite automata, e.g. avoid a fixed pattern, avoid several patterns at once, stay in a horizontal strip and many others more. However, they only considered walks with steps of length one. In this paper we will generalize their results to walks with longer steps. We will also give some applications of this extension and prove a conjecture about the asymptotic behavior of the expected number of ascents in Schroeder paths.