On Parikh Matrices

Mateescu et al (2000) introduced an interesting new tool, called Parikh matrix, to study in terms of subwords, the numerical properties of words over an alphabet. The Parikh matrix gives more information than the well-known Parikh vector of a word which counts only occurrences of symbols in a word. In this note a property of two words u, v, called “ratio property”, is introduced. This property is a sufficient condition for the words uv and vu to have the same Parikh matrix. Thus the ratio property gives information on the M−ambiguity of certain words and certain sets of words. In fact certain regular, context-free and context-sensitive languages that have the same set of Parikh matrices are exhibited. In the study of fair words, Cerny (2006) introduced another kind of matrix, called the p−matrix of a word. Here a “weak-ratio property” of two words u, v is introduced. This property is a sufficient condition for the words uv and vu to have the same p−matrix. Also the words uv and vu are fair whenever u, v are fair and have the weak ratio property.