Parikh matrices of arrays under Dejean array morphism

Abstract

The Parikh matrix of a word w over an ordered alphabet is an upper triangular matrix associated with the word w. The entries of this matrix above the main diagonal are numbers of certain subsequences, called subwords in the word w. Properties of Parikh matrices of morphic images of words under different types of morphisms have also been investigated. On the other hand, a rectangular picture array of symbols is an extension of the notion of a word to two dimensions. The concept of Parikh matrix was extended to such arrays by defining two kinds of matrices, called row and column Parikh matrices and several properties have been established. Here we introduce an array morphism, called Dejean array morphism defined on three symbols. This array morphism extends to two dimensions a word morphism due to Dejean. We derive formulae to count subwords in the rows of image arrays of rectangular binary picture arrays under Dejean array morphism, thus yielding the row Parikh matrix of the image array.