Enumeration of paths in a hexagonal circle packing

TL;DR

This paper systematically enumerates paths in a hexagonal circle packing using advanced combinatorial methods, providing explicit formulas, bijections, and asymptotic analysis to deepen understanding of geometric and combinatorial properties.

Contribution

It introduces novel enumeration techniques for paths in hexagonal packings, including generating functions, bijections with known path classes, and continued-fraction expansions.

Findings

Closed-form bivariate generating functions derived

Bijections established with skew Dyck and Motzkin paths

Asymptotic formulas and Riordan array structures identified

Abstract

We investigate paths in the hexagonal circle packing and enumerate them with respect to width, height, number of steps, area, and kissing number. Functional equations and the kernel method yield closed bivariate generating functions together with coefficient formulas and asymptotics. We establish bijections with skew Dyck paths, constrained Motzkin paths, and peakless Motzkin paths, and show that several of the associated counting arrays are Riordan arrays. Continued-fraction expansions for the area and kissing-number enumerators are also obtained.