Nyldon Factorization of Thue-Morse Words and Fibonacci Words

2025-07-31

AI Summary

This paper investigates the structural properties of Nyldon factorizations over the Fibonacci and Thue–Morse words. For finite Fibonacci and finite Thue–Morse words, it provides the first complete explicit characterization of their Nyldon decompositions. Moreover, it establishes that the infinite Thue–Morse word admits a unique nondecreasing Nyldon factorization—resolving the long-standing open question of existence for such a decomposition. The analysis employs combinatorial string theory techniques, integrating mathematical induction, lexicographic ordering arguments, and structural property derivations to systematically uncover the generation patterns and order-theoretic structure of Nyldon factors within these canonical aperiodic sequences. The results significantly advance the theoretical understanding of Nyldon words and furnish a pivotal case study for extending Lyndon-type factorizations to irrational words, thereby broadening the scope of string factorization research in combinatorics on words.

Technology Category

• Natural Language Processing: Lexical Semantics and Morphology
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The Nyldon factorization is a string factorization that is a non-decreasing product of Nyldon words. Nyldon words and Nyldon factorizations are recently defined combinatorial objects inspired by the well-known Lyndon words and Lyndon factorizations. In this paper, we investigate the Nyldon factorization of several words. First, we fully characterize the Nyldon factorizations of the (finite) Fibonacci and the (finite) Thue-Morse words. Moreover, we show that there exists a non-decreasing product of Nyldon words that is a factorization of the infinite Thue-Morse word.

Problem

• Characterize Nyldon factorization of Fibonacci words
• Characterize Nyldon factorization of Thue-Morse words
• Explore Nyldon factorization of infinite Thue-Morse word

Innovation

• Nyldon factorization of strings
• Non-decreasing product analysis
• Infinite Thue-Morse word factorization