The vectorial kernel method for walks with longer steps

Abstract

Asinowski, Bacher, Banderier and Gittenberger [1] recently developed the vectorial kernel method – a powerful extension of the classical kernel method useable 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 Schröder paths.

Mathematics Subject Classification (2020):

• 05A15
• 05A16

Key words:

• Lattice path
• Schröder path
• generating functions
• kernel method
• asymptotic behavior