Two-Dimensional Digitized Picture Arrays and Parikh Matrices

Parikh matrix mapping or Parikh matrix of a word has been introduced in the literature to count the scattered subwords in the word. Several properties of a Parikh matrix have been extensively investigated. A picture array is a two-dimensional connected digitized rectangular array consisting of a finite number of pixels with each pixel in a cell having a label from a finite alphabet. Here we extend the notion of Parikh matrix of a word to a picture array and associate with it two kinds of Parikh matrices, called row Parikh matrix and column Parikh matrix. Two picture arrays A and B are defined to be M-equivalent if their row Parikh matrices are the same and their column Parikh matrices are the same. This enables to extend the notion of M-ambiguity to a picture array. In the binary and ternary cases, conditions that ensure M-ambiguity are then obtained

Algebraic Properties of Parikh Matrices of Binary Picture Arrays

A word is a finite sequence of symbols. Parikh matrix of a word is an upper triangular matrix with ones in the main diagonal and non-negative integers above the main diagonal which are counts of certain scattered subwords in the word. On the other hand a picture array, which is a rectangular arrangement of symbols, is an extension of the notion of word to two dimensions. Parikh matrices associated with a picture array have been introduced and their properties have been studied. Here we obtain certain algebraic properties of Parikh matrices of binary picture arrays based on the notions of power, fairness and a restricted shuffle operator extending the corresponding notions studied in the case of words. We also obtain properties of Parikh matrices of arrays formed by certain geometric operations