Parikh Matrices, Amiability and Istrail Morphism

Abstract

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.

On Core Words and the Parikh Matrix Mapping

“…If two words are M -equivalent, they are also said to be amiable as in [2][3][4]. Note that the notion of M -equivalence depends on the ordered alphabet Σ.…”

Morphic images of binary words and Parikh matrices

“…Parikh mapping has been extended in [5] by introducing the concept of a Parikh matrix mapping or Parikh matrix, which gives more numerical information about a word w by counting certain subwords (also called scattered subwords) in the word. Intensive investigations [See, for example, [6][7][8][9][10][11][12][13][14] have been done on several properties of Parikh matrices, such as M-ambiguity, injectivity, commutativity and so on, leading to interesting results as well as questions that remain unsolved.…”

“…A mapping on words w whose images (w) are also words, is called a morphism if satisfies the property that (uv) = (u) (v) for given words u, v. Using a specific kind of morphism, called Istrail morphism [15] on a set {a,b,c} of three symbols, Atanasiu [12] investigated, based on Parikh matrices, the property of M-ambiguity or amiability of morphic images of words under the Istrail morphism. Parikh matrices of words that involve certain ratio-property, called weak-ratio property, are investigated by Subramanian et al [10].…”

Power Sums Associated With Certain Recursive Procedures on Words

“…. , i t ) of increasing positive integers, where for 1 ≤ j ≤ t, the jth letter of u is the i j th letter of w. For instance, the five occurrences of u = aaba in w = abababab are (1,3,4,5), (1,3,4,7), (1,3,6,7), (1,5,6,7), (3,5,6,7).…”

“…(Such words have also been called amiable, for instance, in [1], [2].) It is also clear that, for words over a k-letter alphabet, k-equivalence implies matrix equivalence.…”