Combinatorics of generalized Dyck and Motzkin paths

TL;DR

This paper explores the combinatorial structures of generalized Dyck and Motzkin paths, linking them to generalized exclusion statistics and deriving explicit counting formulas for paths with fixed step distributions.

Contribution

It introduces a novel connection between path combinatorics and exclusion statistics, providing explicit enumeration formulas and a new class of generalized compositions.

Findings

Derived explicit formulas for counting paths with fixed steps at each level.

Established a link between path combinatorics and generalized exclusion statistics.

Identified a new class of generalized compositions related to path length.

Abstract

We relate the combinatorics of periodic generalized Dyck and Motzkin paths to the cluster coefficients of particles obeying generalized exclusion statistics, and obtain explicit expressions for the counting of paths with a fixed number of steps of each kind at each vertical coordinate. A class of generalized compositions of the integer path length emerges in the analysis.