Rich, Sturmian, and trapezoidal words

TL;DR

This paper investigates the relationships between rich, Sturmian, and trapezoidal words, revealing that trapezoidal palindromes are precisely Sturmian and characterizing Sturmian and rich palindromes through complexity measures.

Contribution

It establishes new characterizations of Sturmian and rich palindromes based on subword and palindromic complexity, clarifying their interconnections.

Findings

Trapezoidal palindromes are exactly Sturmian.

Sturmian palindromes can be characterized by their complexity measures.

Rich palindromes relate to subword and palindromic complexity.

Abstract

In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced in arXiv:0801.1656 by the second and third authors together with J. Justin and S. Widmer, constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors. Every finite Sturmian word is rich, but not conversely. Trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite Sturmian words. Unfortunately this property does not characterize finite Sturmian words. In this note we show that the only trapezoidal palindromes are Sturmian. More generally we show that Sturmian palindromes can be characterized either in terms of their subword complexity (the trapezoidal property) or in terms of their palindromic complexity. We also obtain a similar characterization of rich palindromes in terms of a relation between palindromic complexity and subword complexity.