Certain upper triangular matrices, termed as Parikh matrices, are often used in the combinatorial study of words. Given a word, the Parikh matrix of that word elegantly computes the number of occurrences of certain predefined subwords in that word. In this paper, we compute the Parikh matrix of any word raised to an arbitrary power. Furthermore, we propose canonical decompositions of both Parikh matrices and words into normal forms. Finally, given a Parikh matrix, the relation between its normal form and the normal forms of words in the corresponding M-equivalence class is established.

中文翻译：

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词的幂的 Parikh 矩阵

某些上三角矩阵，称为 Parikh 矩阵，经常用于单词的组合研究。给定一个词，该词的 Parikh 矩阵优雅地计算该词中某些预定义子词的出现次数。在本文中，我们计算任何单词的任意幂的 Parikh 矩阵。此外，我们建议将 Parikh 矩阵和单词规范分解为规范形式。最后，给定一个 Parikh 矩阵，建立其范式与对应 M 等价类中词的范式之间的关系。