A Classification of Trapezoidal Words

TL;DR

This paper classifies trapezoidal words, showing that open ones are primitive and closed ones are Sturmian, and explores properties of trapezoidal palindromes and their relation to Sturmian words.

Contribution

It introduces a classification of trapezoidal words into open and closed, linking closed trapezoidal words to Sturmian words and analyzing their properties.

Findings

Open trapezoidal words are primitive.

Closed trapezoidal words are Sturmian.

Trapezoidal palindromes are closed and Sturmian.

Abstract

Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A trapezoidal word is closed if its longest repeated prefix has exactly two occurrences in the word, the second one being a suffix of the word. Otherwise it is open. We show that open trapezoidal words are all primitive and that closed trapezoidal words are all Sturmian. We then show that trapezoidal palindromes are closed (and therefore Sturmian). This allows us to characterize the special factors of Sturmian palindromes. We end with several open problems.