Reducing the ambiguity of Parikh matrices

The Parikh matrix mapping allows us to describe words using matrices. Whilst compact, this description comes with a level of ambiguity since a single matrix may describe multiple words. In this paper, we investigate how considering the Parikh matrices of various transformations of a given word can decrease that ambiguity. More specifically, for any word, we study the Parikh matrix of its projection to a smaller alphabet as well as that of its Lyndon conjugate. Our results demonstrate that ambiguity can often be reduced using these concepts, and we give conditions on when they succeed.