# Palindromes in two-dimensional Words

## TL;DR

This paper explores the properties of two-dimensional palindromes, focusing on HV-palindromes, and proves a conjecture about their maximum number of distinct sub-arrays, also analyzing their minimal occurrence in infinite words.

## Contribution

It introduces structural properties of HV-palindromes, proves a conjecture on their maximum count in finite words, and determines the minimal number in infinite words over finite alphabets.

## Key findings

- Proved the maximum number of HV-palindromic sub-arrays in finite 2D words.
- Established the least number of HV-palindromes in infinite 2D words.
- Compared properties of 2D palindromes and HV-palindromes.

## Abstract

A two-dimensional ($2$D) word is a $2$D palindrome if it is equal to its reverse and it is an HV-palindrome if all its columns and rows are $1$D palindromes. We study some combinatorial and structural properties of HV-palindromes and its comparison with $2$D palindromes. We investigate the maximum number number of distinct non-empty HV-palindromic sub-arrays in any finite $2$D word, thus, proving the conjecture given by Anisiua et al. We also find the least number of HV-palindromes in an infinite $2$D word over a finite alphabet size $q$.