Novel Constructions of Words with Strong Avoidance Properties and their Combinatorial Analysis

TL;DR

This paper introduces new mathematical constructions of infinite words with strong pattern avoidance properties, providing theoretical results and conjectures that deepen understanding in combinatorics on words and symbolic dynamics.

Contribution

It presents novel constructions of strongly pattern-avoiding words using cyclic shift morphisms and establishes their avoidance properties with a proof, along with a conjecture on their factor complexity.

Findings

Defined Strongly (k, δ)-Free Words with cyclic shift morphisms

Proved avoidance properties of these words

Proposed a conjecture on their factor complexity

Abstract

This paper begins with a comprehensive overview of combinatorics on words and symbolic dynamics, covering their historical origins, fundamental concepts, and interconnections. Building upon this foundation, we introduce novel mathematical constructions related to pattern avoidance in infinite words. Specifically, we define Strongly $(k, \delta)$-Free Words generated via cyclic shift morphisms and present a theorem establishing specific avoidance properties for these words, along with a detailed proof. Furthermore, we propose a conjecture regarding their factor complexity. These original results contribute to the theoretical understanding of word structures and their combinatorial properties, opening avenues for further research in discrete mathematics.