Parikh Motivated Study on Repetitions in Words

TL;DR

This paper explores the structure of words through canonical decompositions, studies square-free ternary words via Parikh matrices, and constructs infinite classes of such words sharing the same Parikh matrix.

Contribution

It introduces the concept of general prints and core prints of words, analyzes their properties, and extends the study of square-free words using Parikh matrix mapping and morphisms.

Findings

Finite number of matrix-equivalence classes of square-free ternary words.

Existence of infinitely many square-free ternary word pairs with identical Parikh matrices.

Characterization of core prints and their role in word repetition analysis.

Abstract

We introduce the notion of general prints of a word, which is substantialized by certain canonical decompositions, to study repetition in words. These associated decompositions, when applied recursively on a word, result in what we term as core prints of the word. The length of the path to attain a core print of a general word is scrutinized. This paper also studies the class of square-free ternary words with respect to the Parikh matrix mapping, which is an extension of the classical Parikh mapping. It is shown that there are only finitely many matrix-equivalence classes of ternary words such that all words in each class are square-free. Finally, we employ square-free morphisms to generate infinitely many pairs of square-free ternary words that share the same Parikh matrix.