Factor frequencies in generalized Thue-Morse words

TL;DR

This paper investigates the frequencies of factors in generalized Thue-Morse words, extending previous work by analyzing their structure and applying known results to determine factor distributions.

Contribution

It provides a detailed analysis of factor frequencies in generalized Thue-Morse words using morphism-based definitions and existing theoretical frameworks.

Findings

Explicit formulas for factor frequencies in generalized Thue-Morse words

Extension of previous results to a broader class of morphic words

Insights into the combinatorial structure of these words

Abstract

We describe factor frequencies of the generalized Thue-Morse word t_{b,m} defined for integers b greater than 1, m greater than 0 as the fixed point starting in 0 of the morphism \phi_{b,m} given by \phi_{b,m}(k)=k(k+1)...(k+b-1), where k = 0,1,..., m-1 and where the letters are expressed modulo m. We use the result of A. Frid, On the frequency of factors in a D0L word, Journal of Automata, Languages and Combinatorics 3 (1998), 29-41 and the study of generalized Thue-Morse words by S. Starosta, Generalized Thue-Morse words and palindromic richness, arXiv:1104.2476v2 [math.CO].