The injectivity of Parikh mapping and Parikh matrix mapping over a binary alphabet

Abstract

The Parikh vector is an old and vitally important within the idea of Formal languages. The Parikh vector of a word counts the number of occurrences of each member of an alphabet. Siromoney et al introduced the Generalization of the Parikh vector (GPV) which indicates the position of each member of the alphabet. The Parikh mapping is defined for finite words where GPV is defined for infinite words also. Mateescu et al introduced a new tool called Parikh matrix which is the extension of Parikh vector and the concept is to study numerical properties of words over an alphabet, in terms of sub words. In this field of research work, sub word refers both continues sequence of members and also scattered sequence of members. Amiable words are those that are defined by having the same corresponding Parikh matrix mapping. In this paper injectivity of Parikh mapping and Parikh matrix mapping on two famous infinite sequences are studied and new properties are obtained under the concepts of amiable.