with Joonghyuk Hahn, Yo-Sub Han, and Kai Salomaa. (DLT'24)

The Parikh matrix is a morphism from the monoid of words to matrices. Unlike Parikh vector (i.e. abelianization) which captures how many times specific character has appeared in a given word, Parikh matrix counts more sophisticated combinatorial patterns of the word. Since the introduction of the Parikh matrix, how sharp the Parikh matrix identifies the given word was considered as a major research interest. 

This paper provides the first general characterization of when two words have the same Parikh matrix, which has been open for more than 20 years. Previously, the characterization was known only for the alphabet of size 2 or 3. The extended version of this paper is under submission.