Criteria for the matrix equivalence of words

In the algebraic study of words and languages it is often convenient to analyze words as numerical quantities. One such now-famous example is the study of words by using their attached Parikh mapping (or vector). However, the Parikh vector abstracts away too much of the structure of the word. Parikh matrices were introduced as a means to obtain more than just the number of occurrences of single letters. When studying properties of words, an important property is that a word is uniquely determined by the number of occurrences of certain predetermined subwords. In the context of Parikh matrices, the problem is translated in finding which words are completely characterized by their associated Parikh matrix. This paper links this problem with that of the print of a word, i.e. the word obtained by considering consecutive occurrences of the same letter as only one letter. We obtain results regarding finiteness and context-freeness of such classes of words.

On parikh matrices, ambiguity, and prints

Independence of certain quantities indicating subword occurrences

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.

On Core Words and the Parikh Matrix Mapping

Core words and Parikh matrices

We deal with the notion of M-unambiguity [5] in connection with the Parikh matrix mapping introduced by Mateescu and others in [7]. M-unambiguity is studied both in terms of words and matrices and several sufficient criteria for M-unambiguity are provided in both cases, nontrivially generalizing the criteria based on the γ-property introduced by Salomaa in [15]. Also, the notion of M-unambiguity with respect to a word is defined in connection with the extended Parikh matrix morphism [16] and some of the M-unambiguity criteria are lifted from the classical setting to the extended one. 

This paper is a revised and extended version of [17].

Virgil Nicolae Şerbănuţă

Papers

Injectivity of the Parikh Matrix Mappings Revisited