Generalized trapezoidal words

TL;DR

This paper introduces generalized trapezoidal words over any finite alphabet, explores their properties, and characterizes which are rich in palindromes, extending the concept beyond binary alphabets.

Contribution

It generalizes trapezoidal words to arbitrary alphabets and analyzes their combinatorial and structural properties, including palindrome richness.

Findings

Not all GT-words are rich in palindromes for alphabets of size three or more

Characterization of rich GT-words over larger alphabets

Extension of trapezoidal word theory beyond binary cases

Abstract

The factor complexity function $C_w(n)$ of a finite or infinite word $w$ counts the number of distinct factors of $w$ of length $n$ for each $n \ge 0$. A finite word $w$ of length $|w|$ is said to be trapezoidal if the graph of its factor complexity $C_w(n)$ as a function of $n$ (for $0 \leq n \leq |w|$) is that of a regular trapezoid (or possibly an isosceles triangle); that is, $C_w(n)$ increases by 1 with each $n$ on some interval of length $r$, then $C_w(n)$ is constant on some interval of length $s$, and finally $C_w(n)$ decreases by 1 with each $n$ on an interval of the same length $r$. Necessarily $C_w(1)=2$ (since there is one factor of length $0$, namely the empty word), so any trapezoidal word is on a binary alphabet. Trapezoidal words were first introduced by de Luca (1999) when studying the behaviour of the factor complexity of finite Sturmian words, i.e., factors of infinite "cutting sequences", obtained by coding the sequence of cuts in an integer lattice over the positive quadrant of $\mathbb{R}^2$ made by a line of irrational slope. Every finite Sturmian word is trapezoidal, but not conversely. However, both families of words (trapezoidal and Sturmian) are special classes of so-called "rich words" (also known as "full words") - a wider family of finite and infinite words characterized by containing the maximal number of palindromes - studied in depth by the first author and others in 2009. In this paper, we introduce a natural generalization of trapezoidal words over an arbitrary finite alphabet $\mathcal{A}$, called generalized trapezoidal words (or GT-words for short). In particular, we study combinatorial and structural properties of this new class of words, and we show that, unlike the binary case, not all GT-words are rich in palindromes when $|\mathcal{A}| \geq 3$, but we can describe all those that are rich.