Partial skew Dyck paths -- a kernel method approach

TL;DR

This paper studies partial skew Dyck paths and their variants using generating functions and the kernel method, exploring different path constraints and symmetries to derive explicit formulas and analyze their combinatorial properties.

Contribution

It introduces a kernel method approach to analyze partial skew Dyck paths and their decorated and dual variants, providing new explicit generating function formulas.

Findings

Derived explicit generating functions for partial skew Dyck paths.

Analyzed dual path versions with reversed reading order.

Explored paths with negative territory, revealing increased computational complexity.

Abstract

Skew Dyck are a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ a south-west step $(-1,-1)$ is also allowed, provided that the path does not intersect itself. Replacing the south-west step by a red south-east step, we end with decorated Dyck paths. We analyze partial versions of them where the path ends on a fixed level $j$, not necessarily at level 0. We exclusively use generating functions and derive them with the celebrated kernel method. In the second part of the paper, a dual version is studied, where the paths are read from right to left. In this way, we have two types of up-steps, not two types of down-steps, as before. A last section deals with the variation that the negative territory (below the $x$-axis) is also allowed. Surprisingly, this is more involved in terms of computations.