No abstract

Using the fact that the Parikh matrix mapping is not an injective mapping, the paper investigates some properties of the set of the binary words having the same Parikh matrix; these words are called "amiable" Some results concerning the conditions when the equivalence classes of amiable words have more than one element, a characterization theorem concerning a graph associated to an equivalence class of amiable words, and some basic properties of a rank distance defined on these classes are the main subjects considered here.

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.

Using the fact that the Parikh matrix mapping is not an injective mapping, the paper investigates some properties of the set of words having the same Parikh matrix; these words are called "amiable" or "M - equivalent". The presented paper uses the results obtained in [3] for the binary case. The aim is to distinguish the amiable words by using a morphism that provides additional information about them. The morphism proposed here is the Istrail morphism.