Enumeration and Structure of Trapezoidal Words

TL;DR

This paper characterizes trapezoidal words, provides their enumeration, and explores their classification into open and closed types, revealing connections with Sturmian palindromes, semicentral words, and Fibonacci word prefixes.

Contribution

It introduces combinatorial characterizations of trapezoidal words, formulas for their enumeration, and a novel classification into open and closed, linking them to Sturmian palindromes and semicentral words.

Findings

Sturmian palindromes are closed trapezoidal words.

A closed trapezoidal word is a Sturmian palindrome iff its longest repeated prefix is a palindrome.

The sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence.

Abstract

Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated prefix is a palindrome. We also define a new class of words, \emph{semicentral words}, and show that they are characterized by the property that they can be written as $uxyu$, for a central word $u$ and two different letters $x,y$. Finally, we investigate the prefixes of the Fibonacci word with respect to the property of being open or closed trapezoidal words, and show that the sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence.