Parikh Matrices and Istrail Morphism

A word wwis a sequence of symbols. A scattered subword or simply a subword uuof the word wwis a subsequence of ww. Parikh matrix MM(ww)is an ingenius tool introduced by Mateescu et al (2001) to count certain subwords in a word ww. Various properties of Parikh matrices have been established. Two words uuand vv are said to be M-ambiguous or amiable if their Parikh matrices MM(uu) and MM(vv)are the same. On the other hand a morphism ffis a mapping on words wwwhose images ff(ww)are also words with the property that,ff(uuvv) = ff(uu)ff(vv)for given words uu and vv. Istrailmorphism (Istrail, 1977) is a specific kind of morphism on a set {aa,bb, cc}of three symbols. Using this morphism, M-ambiguity or amiability of words based on Parikh matrices is investigated by Atanasiu (2010). Parikh matrices of words that involve certain ratio-property are investigated by Subramanian et al (2009). Here we consider this kind of ratio-property in the context of Istrailmorphism and obtain certain properties of morphic images of words under Istrailmorphism. Using these properties,conditions are obtained for product of Parikh matrices of such morphic images under Istrailmorphismto commute. | Combinatorics | Word | Subword | Scattered Subword | Parikh Matrix | Morphism | ® 2013 IbnuSina Institute. All rights reserved. http://dx.doi.org/10.11113/mjfas.v9n1.72


INTRODUCTION
Combinatorics on words is a growing area of discrete mathematics with applications in many different fields.An excellent sourcefor the study of combinatorial problems on words and their applications is the reference [1].The topic ofcombinatorics on words deals with general properties of words.A word is a finite or an infinite sequence of symbols of a finite set called an alphabet.For example,  is a word over the alphabet {, }, which has an additional property of being a palindrome.Investigation ofproperties of words has given rise to challenging problems.
Studies on numerical properties of words have proved to be useful.Parikh mapping or Parikh vector [2] is one such numerical property, which counts the number of occurrences of each symbol in a word over an alphabet.For example, the Parikh vector of the word is (3,4), since it has three a's and four b's.Thishas been an important notion in formal language theory [3,4], which is a branch of theoretical computer science.
Recently, Parikh mapping has been extended in [5]by introducing the concept of a Parikh matrix mapping or Parikh matrix, which gives more numerical information than a Parikh vector does.In fact Parikh matrixM(w)counts certain subwords in a wordw, also called scattered subwords.Since the introduction of the notion of Parikh matrix, intensive investigations have been done on several problems related to subwords [6][7][8][9][10][11][12][13].Among various properties of Parikh matrices that have been established, Mambiguity or amiability of words has been extensively investigated.Two words  and are said to be Mambiguous or amiable if their Parikh matrices are the same, that is () = ().
On the other hand a morphismis a mapping on words w whose images()are also words with the property that () = ()(), for given words  and .Istrailmorphism [14]is a specific kind of morphism on a set {, , }of three symbols.M-ambiguity or amiability of words based on Parikh matrices is recently investigated by Atanasiu [13]using this morphism.
Parikh matrices of words that involve certain ratioproperty are investigated by Subramanian et al [9].This property has been used in [12] to obtain conditions for equality of Parikh matrices of products  and  of two words  and in the case of binary and three-symbol alphabets.
Here we considermorphic images of words under Istrailmorphismand obtain properties of these morphic images in the context of Istrailmorphism, using this kind of ratio-property.In particular certain conditions for commutativity of product of Parikh matrices of such morphic images under Istrailmorphism are obtained.

PRELIMINARIES
An alphabetis a finite set of symbols.A finite sequence of symbols from is called a finite word or simply a word over .The set of all words over is denoted by * .The empty word with no symbols is denoted by .
The length of a word  ∈  * is the number of symbols (counting repetitions) in the word  and is denoted by||.
A word  is called a subword (also called scatteredsubword) of a word  if there exist words 1 , … ,   and  0 , … ,   , such that  =  1 …   and  =  0  1  1 …     .For example if  =  is a word over the alphabet {, }, the symbols in the positions 2, 5, 6, 7, 9, namely,  is a subword of .The number of occurrences of the word  as a subword of the word  is denoted by ||  .
The notion of a Parikh matrix introduced in [5], is an extension of the notion of Parikh vector and gives more numerical information than a Parikh vector does.In fact Parikh matrix counts the number of occurrences of certain subwords in a given word.In the rest of the paper, we are concerned mainly with words over a binary ordered alphabet{ < } or a three-symbol ordered alphabet { <  < }.
Let   denote the set of all ×  upper triangular matrices M such that M has no negative integer entry, has 1's in the main diagonal and has 0's for all entries below the main diagonal.
Let  = { < }.The Parikh matrix mapping  2 is a mapping from  * to  3 , given by Note that the notion of Parikh matrix mapping can be defined for words over a general ordered alphabet Σ = { 1 <  2 < ⋯ <   }.But we have recalled here the definition in the case of a binary alphabet.For a threesymbol alphabetΣ = { <  < }, the Parikh matrix mapping  3 is a mapping from  * to  4 , given by As before, for a word  ∈  * , 3 () is obtained by substituting for each symbol a, b or c in w, the corresponding matrix Ψ 3 () ,Ψ 3 () or Ψ 3 () and by performing matrix multiplication.
We illustrate with examples.

Example 1
Note that the word  =  has threea's,threeb's and six subwordab's.The Parikh vector (3,3) appears at the second diagonal above the main diagonal in the Parikh matrix.
M-ambiguity of words has been extensively investigated.Characterizations of M-ambiguous binary words have been established.We state here one such characterization [8].
Two binary words u, v over the alphabet{ < } have the same Parikh matrix and hence M-ambiguous if and only if v can be obtained from u by interchanging an occurrence of  in  with a distinct occurrence of in .It is known that such a characterization does not immediately carry over to words over a three-symbol alphabet.
We also denote the Parikh matrix of a word w by ().
While investigating the notion of M-ambiguity of words over a three-symbol ordered alphabet { <  < } , a property, called weak-ratio property was introduced in [9].Two words u, v over  = { <  < } are said to satisfy the weak-ratio property, written  ~  if Weak-ratio property of binary words u,v over{ <  can be similarly defined by requiring It is shown in [9]that weak-ratio property of words u, v is a sufficient condition for equality of the Parikh matrices of the product binary words uv, vu or in other words, () = ().
Recently, Atanasiu [13]studied M-ambiguity of words that are morphic images of binary words under a special kind of morphism known as Istrailmorphism [14].If A and B are two finite nonempty alphabets, a morphism is a mapping:  * →  * such that () = ()() for all ,  ∈  * .

MORPHIC IMAGES UNDER ISTRAIL MORPHISM
Here we consider the weak-ratio property [9]of words and obtain first some properties of the morphic images under Istrailmorphism,restricted to words over a binary alphabet{ < }.Note that the words () are over a three-symbol alphabet { <  < }.

Theorem 1
Let  = { < }.Then we have the following properties: But ()has as many b's as whas a's.i.e.
From property (i), we have Note that the application of  on  yields the factor  for every in  and so the contribution coming from these factors to the number of subword in () is ||  +(||  − 1) + ⋯ + 1 whereas the application of  on  yields the factor  for every  in u while removing the b and so the contribution coming from this to the number of subwordin () is the same as the number of subword in u.This proves i).

|𝜑𝜑(𝑢𝑢)|
This proves (ii).∎ Parikh matrices of products uv and vu ofwords u, v and their commutativity have been studied in [12].We state a result in [12]on the commutativity of such products in the following Lemma, which we need to prove a result in the subsequent theorem on Parikh matrices of products of morphic images of two words under the Istrailmorphism.Also from Theorem 1, ()~  () since ~   and using Lemma 1, we have is obtained by substituting for each symbol a or b in w, the corresponding matrix Ψ 2 () or Ψ 2 () and by performing matrix multiplication.The matrix  2 () is called the Parikh matrix of the wordw.Note that ||  , ||  , ||  give the number of a's, the number of b's and the number of subword  in .

𝛹𝛹 3 (
) =  3 () =  3 () 3 () 3 () 3 () 3 () 3 () 3 ()In the word w, there are 3 a's, 2 b's, 2 c's, 5 ab's, 3 bc's, 7 abc's.The entries in the Parikh matrix of , in the second diagonal (above the main diagonal) give the number of a's, b's, c's, in ; the entries in the diagonal above it give the number of subwordsab's, bc's in ; and the rightmost entry in the first row gives the number of subwordabc's in w.The Parikh matrix is known[5,7]to be not injective which means that two or more words can have the same Parikh matrix.For example the following five words over the alphabet  = { < } , , , ,  have the same Parikh matrix � Words having the same Parikh matrix are called Mequivalent or amiable.