Enumeration of \L{}ukasiewicz paths modulo some patterns

TL;DR

This paper classifies ukasiewicz paths based on pattern occurrence positions, providing enumeration results and a bijection with Motzkin paths for certain classes, advancing combinatorial understanding.

Contribution

It introduces a novel enumeration of ukasiewicz paths modulo pattern occurrence and establishes a bijection with Motzkin paths for specific classes.

Findings

Enumeration formulas for equivalence classes of ukasiewicz paths

A constructive bijection between Motzkin paths and ukasiewicz path classes

Insights into pattern occurrence structures in lattice paths

Abstract

For any pattern $\alpha$ of length at most two, we enumerate equivalence classes of \L{}ukasiewicz paths of length $n\geq 0$ where two paths are equivalent whenever the occurrence positions of $\alpha$ are identical on these paths. As a byproduct, we give a constructive bijection between Motzkin paths and some equivalence classes of \L{}ukasiewicz paths.