Parikh Matrices for Powers of Words

TL;DR

This paper explores the properties of Parikh matrices related to powers of words, introduces canonical decompositions, and establishes relations between normal forms and M-equivalence classes.

Contribution

It provides a method to compute Parikh matrices for word powers and introduces normal forms for matrices and words, enhancing combinatorial analysis tools.

Findings

Computed Parikh matrices for arbitrary word powers

Proposed canonical decompositions and normal forms

Established relations between matrix and word normal forms

Abstract

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.