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 Şerbănuţă. Recently, a decade-old conjecture by Şerbǎnuţǎ on the M-ambiguity of words has 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 each of those words is 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 result in an M-ambiguity sequence that is ultimately periodic.

中文翻译：

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Parikh矩阵的M-歧义序列及其周期性

Mateescu等人介绍了Parikh矩阵。在2001年，各种研究者引发了对形式语言理论的许多新研究，其中包括Şerbănuţă。最近，Şerbǎnuţǎ关于单词M-歧义性的十年猜想得到了证明，这为研究此类单词带来了新的可能性。在本文中，我们研究了单词中字母的选择性重复重复如何影响结果单词的M模糊度。然后，将这些单词中每个单词的相应M-模糊度按序列表示，我们称之为M-模糊度序列。我们证明了几乎所有的M模式-模糊序列是可以达到的。最后，通过在整数编程中采用某些代数方法和一些基础理论，我们证明了单词中相同类型字母的重复周期性重复会导致最终产生M歧义序列。