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 nonnegative 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 a 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.

中文翻译：

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二元图像阵列Parikh矩阵的代数性质

单词是符号的有限序列。单词的Parikh矩阵是上三角矩阵，在主对角线上有一个，在主对角线上有非负整数，它们是单词中某些分散的子词的计数。另一方面，图片阵列是符号的矩形排列，是单词概念到二维的扩展。已经介绍了与图片阵列相关的Parikh矩阵，并研究了它们的特性。在这里，我们基于幂，公平性的概念，以及扩展了在单词情况下研究的相应概念的受限混洗算符，获得了二进制图片阵列的Parikh矩阵的某些代数性质。我们还获得了由某些几何运算形成的阵列的Parikh矩阵的属性。