Dyck path factorization

TL;DR

This paper introduces a novel method for uniquely decomposing Dyck paths into prime fragments using peaks and valleys, providing a new perspective on their structure and an accompanying Python tool.

Contribution

It presents a new factorization technique for Dyck paths based on peaks and valleys, enabling unique decomposition and analysis.

Findings

Unique factorization of Dyck paths into prime fragments

Characterization of Dyck paths by sets of peaks or valleys

Provision of an online Python tool for factorization

Abstract

Dyck paths are among the most heavily studied Catalan families. We work with peaks and valleys to uniquely decompose Dyck paths into the simplest objects - prime fragments with a single peak. Each Dyck path is uniquely characterized by a set of peaks or a set of valleys. The appendix contains a python program with which the reader can factorize Dyck paths online.