No abstract

The characterization of M-equivalence for the Parikh matrices is a decade old open problem. This paper studies Parikh matrices and M-equivalence in relation to the s-shuffle operator for the binary alphabet. We also study the distance between images under the s-shuffle operator in a graph associated to the corresponding class of M-equivalent words.

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.2000 Mathematics Subject Classification. 68R15, 05A05.

Core of a binary word, recently introduced, is a refined way to characterize binary words having the same Parikh matrices, as well as bridging the connection between binary words and partitions of natural numbers. This paper continues the work by generalizing to higher alphabet. The core of a word as well as the relatived version is the essential part of a word that captures the key information of the word from the perspective of its Parikh matrix. Various nice properties of the cores and some interesting results regarding the M -equivalence classes of ternary words are obtained.

The notion of strong [Formula: see text]-equivalence was introduced as an order-independent alternative to [Formula: see text]-equivalence for Parikh matrices. This paper further studies the notions of [Formula: see text]-equivalence and strong [Formula: see text]-equivalence. Certain structural properties of [Formula: see text]-equivalent ternary words are presented and then employed to (partially) characterize pairs of ternary words that are ME-equivalent (i.e. obtainable from one another by certain elementary transformations). Finally, a sound rewriting system in determining strong [Formula: see text]-equivalence is obtained for the ternary alphabet.