# M-Ambiguity Sequences for Parikh Matrices and Their Periodicity Revisited

## TL;DR

This paper explores how repeated letter duplications affect M-ambiguity in words, demonstrating that most ambiguity patterns are possible and that periodic duplications lead to eventually periodic ambiguity sequences.

## Contribution

It introduces the concept of M-ambiguity sequences and proves their attainability and periodicity properties using algebraic and integer programming methods.

## Key findings

- Nearly all M-ambiguity sequences are attainable.
- Repeated periodic duplications produce eventually periodic sequences.
- Disproved a longstanding conjecture on M-ambiguity.

## Abstract

The introduction of Parikh matrices by Mateescu et al. in 2001 has sparked numerous new investigations in the theory of formal languages by various researchers, among whom is Serbanuta. Recently, a decade-old conjecture by Serbanuta on the M-ambiguity of words was disproved, leading to new possibilities in the study of such words. In this paper, we investigate how selective repeated duplications of letters in a word affect the M-ambiguity of the resulting words. The corresponding M-ambiguity of those words are then presented in sequences, which we term as M-ambiguity sequences. We show that nearly all patterns of M-ambiguity sequences are attainable. Finally, by employing certain algebraic approach and some underlying theory in integer programming, we show that repeated periodic duplications of letters of the same type in a word results in an M-ambiguity sequence that is eventually periodic.