Skew 2-Dyck paths via the kernel method

TL;DR

This paper introduces skew 2-Dyck paths with an added down-step, develops automata and generating functions for their enumeration, and discusses extensions to t-Dyck paths and path scanning methods.

Contribution

It extends the concept of 2-Dyck paths by adding a new step and provides automata and kernel method-based generating functions for enumeration.

Findings

Automaton for skew 2-Dyck paths is constructed.

Generating functions are derived using the kernel method.

Extensions to t-Dyck paths and path scanning approaches are discussed.

Abstract

We continue on a recent concept introduced by Kariuki and Okoth, about skew 2-Dyck paths, introducing an additional down-step $L$, together with the usual steps $U$ (up) and $D$ down. There is the syntactical condition that $UL$ and $LU$ can never occur. An automaton that checks these conditions is introduced, and the relevant generating functions are obtained by applying the kernel method to three functional equations. It is briefly discussed how the setting can be extended to $t$-Dyck paths. As a benefit, prefixes of skew $t$-Dyck paths are also enumerated. An approach that scans 2-Dyck paths from right to left is also discussed.