Languages invariant under more symmetries: overlapping factors versus palindromic richness

TL;DR

This paper explores words invariant under symmetry groups, establishing a stronger inequality relating factor and palindromic complexities, and introduces the concept of G-palindromic richness with examples including the Thue-Morse sequence.

Contribution

It generalizes known complexity inequalities to words invariant under finite symmetry groups and introduces the new concept of G-palindromic richness.

Findings

Proves a stronger inequality for G-invariant words.

Introduces G-palindromic richness as a new property.

Provides examples including the Thue-Morse sequence.

Abstract

Factor complexity $\mathcal{C}$ and palindromic complexity $\mathcal{P}$ of infinite words with language closed under reversal are known to be related by the inequality $\mathcal{P}(n) + \mathcal{P}(n+1) \leq 2 + \mathcal{C}(n+1)-\mathcal{C}(n)$ for any $n\in \mathbb{N}$\,. Word for which the equality is attained for any $n$ is usually called rich in palindromes. In this article we study words whose languages are invariant under a finite group $G$ of symmetries. For such words we prove a stronger version of the above inequality. We introduce notion of $G$-palindromic richness and give several examples of $G$-rich words, including the Thue-Morse sequence as well.