Algebraic Properties of Parikh Matrices of Binary Picture Arrays

Somnath Bera,Linqiang Pan,Atulya K Nagar,Sastha Sriram,K G Subramanian

doi:10.1155/2020/3236405

Abstract

A word is a finite sequence of symbols. Parikh matrix of a word is an upper triangular matrix with ones in the main diagonal and nonnegative integers above the main diagonal which are counts of certain scattered subwords in the word. On the other hand, a picture array, which is a rectangular arrangement of symbols, is an extension of the notion of a word to two dimensions. Parikh matrices associated with a picture array have been introduced, and their properties have been studied. Here, we obtain certain algebraic properties of Parikh matrices of binary picture arrays based on the notions of power, fairness, and a restricted shuffle operator extending the corresponding notions studied in the case of words. We also obtain properties of Parikh matrices of arrays formed by certain geometric operations.

Highlights

“Combinatorics on words” [1] is a comparatively new branch of discrete mathematics with applications in many fields. e work [2] of the Norwegian mathematician Axel ue (1863–1922) is considered to be the origin for the beginning of this new branch of mathematics
A finite word or a word is a finite sequence of symbols in a finite set called an alphabet. e Parikh vector [3] of a finite word, which has played a significant role in the theory of formal languages [3], expresses a numerical property of the word by counting the number of occurrences of the different symbols in the word
E recently introduced notion of the Parikh matrix [4] of a word over an ordered alphabet is an extension of the Parikh vector. e Parikh matrix of a word, which is based on subwords of the word, is a very interesting and effective tool in the study of certain numerical properties of the word

Summary

Introduction

“Combinatorics on words” [1] is a comparatively new branch of discrete mathematics with applications in many fields. e work [2] of the Norwegian mathematician Axel ue (1863–1922) is considered to be the origin for the beginning of this new branch of mathematics. Intensive work (see, for example, [5,6,7,8,9,10,11]) has taken place in investigating properties of words based on associated Parikh matrices. Such theoretical studies have dealt with problems of great interest related to words such as inequalities on the numbers of occurrences of subwords, injectivity of the mapping involved in defining the Parikh matrix, and other directions [12]. E notion of the Parikh matrix of a word has been extended to row and column Parikh matrices of picture arrays in [21], and their properties have been studied. A preliminary version of this work was presented in the conference MICOPAM 2018 [25]

Preliminaries

Fair Picture Arrays

Restricted Shuffle Operator on Picture Arrays

Geometric Operations on Picture Arrays

Concluding Remarks