Ternary is Still Good for Parikh Matrices

TL;DR

This paper investigates Parikh matrices, demonstrating that a 2021 conjecture holds only for ternary alphabets and providing counterexamples for larger alphabets, while also exploring minimal Hamming distances in ternary words.

Contribution

It proves the conjecture by Dick et al. is valid only for ternary alphabets and addresses a question on minimal Hamming distances for ternary word classes.

Findings

The conjecture by Dick et al. holds only for ternary alphabets.

Counterexamples are provided for alphabets larger than three.

The minimal Hamming distance for words sharing a congruency class is characterized for ternary alphabets.

Abstract

The focus of this work is the study of Parikh matrices with emphasis on two concrete problems. In the first part of our presentation we show that a conjecture by Dick at al. in 2021 only stands in the case of ternary alphabets, while providing counterexamples for larger alphabets. In particular, we show that the only type of distinguishability in the case of 3-letter alphabets is the trivial one. The second part of the paper builds on the notion of Parikh matrices for projections of words, discussed initially in this work, and answers, once more in the case of a ternary alphabet, a question posed by Atanasiu et al. in 2022 with regards to the minimal Hamming distance in between words sharing a congruency class.