On the injectivity of the Parikh matrix mapping Adrian Atanasiu ∗ Carlos Mart´ ın-Vide † Alexandru Mateescu ‡ Abstract In this paper we investigate the injectivity of the Parikh matrix mapping. This research is done mainly on the binary alphabet. We identify a family of binary words, refered to as “palindromicly amicable”, such that two such words are palindromicly amicable if and only if they have the same image by the Parikh matrix mapping. Some other related problems are discussed, too. 1 Introduction The Parikh mapping or the Parikh vector is a very important notion in the theory of formal languages. This mapping was introduced in [2]. The main result concerning this mapping is that the image by the Parikh mapping of a context-free language is always a semilinear set. In this paper we investigate the injectivity of the Parikh matrix mapping defined mainly on a binary alphabet. The Parikh matrix mapping is an extension of the Parikh mapping introduced in [1]. The extension is based on a special type of matrices. The classical Parikh vector will appear in such a matrix as the second diagonal. All other entries above the main diagonal contain information about the order of letters in the original word. All matrices are triangular, with 1’s on the main diagonal and 0’s below it. Two words with the same Parikh matrix always have the same Parikh vector, but two words with the same Parikh vector have in many cases different Parikh matrices. Thus, the Parikh matrix gives more information about a word than a Parikh vector. We start with some basic notations and definitions. The set of all positive integers is denoted by N . Let Σ be an alphabet. The set of all words over Σ is Σ ∗ and the * Faculty of Mathematics, Bucharest University, Str. Academiei 14, sector 1, 70109 Bucharest, Romania, email: aadrian@pcnet.ro. Research supported by Spanish Secretar´ ıa de Estado de Edu- caci´ øn, Universidades, Investigaci´ øn y Desarrollo, project SAB1999-0025. † Research Group on Mathematical Linguistics, Rovira i Virgili University, Pl. Imperial T` arraco 1, 43005 Tarragona, Spain, email: cmv@correu.urv.es ‡ Faculty of Mathematics, Bucharest University, Str. Academiei 14, sector 1, 70109 Bucharest, Romania, email: alexmate@pcnet.ro 1