Words with many palindrome pair factors

TL;DR

This paper explores infinite words where factors can be decomposed into two palindromes, showing Sturmian words have this property, while the Thue-Morse word does not, and characterizing certain finite words with maximal palindrome pair factors.

Contribution

It demonstrates that all Sturmian words possess the palindrome pair factor property and characterizes binary words with maximal such factors, expanding understanding of palindrome structures in words.

Findings

All Sturmian words have infinitely many factors as products of two palindromes.

The Thue-Morse word does not have this property.

Characterization of binary words not palindrome pairs but with all proper factors as palindrome pairs.

Abstract

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many n, every length-n factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue-Morse word does not have this property. We investigate finite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair.