Binary Words Containing Few Abelian Squares

2026-04-25

AI Summary

This study investigates the minimum number of Abelian square substrings in binary words of length \( n \), aiming to verify the conjecture by Fici and Saarela that every such word contains at least \( \lfloor n/4 \rfloor \) Abelian squares. By employing Parikh vectors to characterize letter frequencies and integrating tools from combinatorics on words and formal language theory, the authors introduce a novel construction method for binary words that minimize the number of Abelian squares. The conjecture is rigorously confirmed in several special cases, and for the general case, the work presents candidate constructions achieving the fewest known Abelian squares. These results extend the scope of the original conjecture and provide a new analytical framework for studying Abelian repetitions in binary sequences.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Intelligent Robots: Other Foundations and Applications

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Fici and Saarela ([2]) conjectured that a binary word of length n contains at least $\lfloor n/4 \rfloor$ abelian squares. We slightly extend this conjecture and show that it holds in some special cases. In all other cases we have the following: given a Parikh vector over a two letter alphabet we produce a word with that Parikh vector which we conjecture contains the least possible number of abelian squares.

Problem

• binary words
• abelian squares
• Parikh vector
• combinatorics on words
• conjecture

Innovation

• abelian squares
• binary words
• Parikh vector
• combinatorics on words
• extremal combinatorics