Parikh Matrices and Istrail Morphism

A word w is a sequence of symbols. A scattered subword or simply a subword u of the word w is a subsequence of w. Parikh matrix M(w) is an ingenius tool introduced by Mateescu et al (2001) to count certain subwords in a word w. Various properties of Parikh matrices have been established. Two words u and v are said to be M-ambiguous or amiable if their Parikh matrices M(u) and M(v) are the same. On the other hand a morphism f is a mapping on words w whose images f(w) are also words with the property that, f(uv)=f(u)f(v) for given words u and v. Istrail morphism (Istrail, 1977) is a specific kind of morphism on a set {a,b,c} of three symbols. Using this morphism, M-ambiguity or amiability of words based on Parikh matrices is investigated by Atanasiu (2010). Parikh matrices of words that involve certain ratio-property are investigated by Subramanian et al (2009). Here we consider this kind of ratio-property in the context of Istrail morphism and obtain certain properties of morphic images of words under Istrail morphism. Using these properties, conditions are obtained for product of Parikh matrices of such morphic images under Istrail morphism to commute