Algebraic Properties of Parikh Matrices of Words under an Extension of Thue Morphism

The Parikh matrix of a word $w$ over an alphabet $\{a_1, \cdots , a_k \}$ with an ordering $a_1 < a_2 < \cdots a_k,$ gives the number of occurrences of each factor of the word $a_1 \cdots a_k$ as a (scattered) subword of the word $w.$ Two words $u,v$ are said to be $M-$equivalent, if the Parikh matrices of $u$ and $v$ are the same. On the other hand properties of image words under different morphisms have been studied in the context of subwords and Parikh matrices. Here an extension to three letters, introduced by S$\acute{e}\acute{e}$bold (2003), of the well-known Thue morphism on two letters, is considered and properties of Parikh matrices of morphic images of words are investigated. The significance of the contribution is that various classes of binary words are obtained whose images are $M-$equivalent under this extended morphism

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