Novel Constructions of Words with Strong Avoidance Properties and their Combinatorial Analysis
Duaa Abdullah, Jasem Hamoud
TL;DR
The paper addresses constructing infinite words that avoid twisted repetitions defined by permutations, bridging combinatorics on words and symbolic dynamics. It introduces strongly (k,δ)-free words via a delta-twisting operation and analyzes infinite words generated by a cyclic shift morphism. The central result shows that for alphabets of size at least three, with a cyclic permutation and a twisting delta not equal to the first power, the generated word exhibits a strong avoidance property; the proof uses a base case, a synchronization lemma, and infinite descent. A linear factor complexity conjecture is proposed, drawing connections to Sturmian-like sequences and suggesting broad implications for shift spaces and dynamic applications.
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 $(k, δ)$-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.