A new characteristic property of rich words

TL;DR

This paper introduces a new characteristic property of rich words, showing that each factor can be uniquely identified by its longest palindromic prefix and suffix, enhancing understanding of their structure.

Contribution

The paper proves a novel characterization of rich words based on their factors being uniquely determined by their longest palindromic prefix and suffix.

Findings

Rich words are characterized by their maximal number of distinct palindromes.

All complete returns to palindromes are themselves palindromes.

Factors in rich words are uniquely determined by their longest palindromic prefix and suffix.

Abstract

Originally introduced and studied by the third and fourth authors together with J. Justin and S. Widmer in arXiv:0801.1656, rich words constitute a new class of finite and infinite words characterized by containing the maximal number of distinct palindromes. Several characterizations of rich words have already been established. A particularly nice characteristic property is that all 'complete returns' to palindromes are palindromes. In this note, we prove that rich words are also characterized by the property that each factor is uniquely determined by its longest palindromic prefix and its longest palindromic suffix.