The height of skew Dyck paths with two variants of downsteps

TL;DR

This paper studies skew Dyck paths with two types of down-steps, analyzing their height and enumeration using generating functions and kernel methods, revealing that the average height grows proportionally to the square root of the path length.

Contribution

It introduces new enumerative results for skew Dyck paths with two down-step variants, applying generating functions and kernel methods to analyze height and other properties.

Findings

Average height is proportional to the square root of path length

Derived explicit generating functions for these paths

Provided enumeration formulas for height and related statistics

Abstract

Recently, in the context of walks of hexagonal circle packings, interest has emerged in the family of skew Dyck paths with two variants of down-steps. These paths have steps $U, D_g, D_b, L=D_r$. Using generating functions, the kernel method and (in)finite linear systems, contributions to the (average) height and other enumerations are made. As in many similar instances, the average height is of order $\sqrt n$.