A Classification of Trapezoidal Words

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.

💡 Research Summary

The paper investigates the class of trapezoidal words—finite binary words whose factor complexity satisfies f w(n) ≤ n + 1 for every n ≥ 0. While every finite Sturmian word is trapezoidal, the converse does not hold, as illustrated by the non‑Sturmian example “aaabab”. Building on earlier work that characterized non‑Sturmian trapezoidal words via a pair of pathological factors (f, g), the authors introduce a novel dichotomy: closed versus open trapezoidal words.

A word is defined as closed if its longest repeated prefix occurs exactly twice, the second occurrence being a suffix of the word; equivalently, the word is a complete return to that prefix. Otherwise the word is open. This notion coincides with the previously studied “periodic‑like” words.

The main contributions are as follows:

1. Open trapezoidal words are primitive (Lemma 13). The proof shows that if an open trapezoidal word were a non‑trivial power, its longest repeated prefix would have more than two occurrences, contradicting the definition of openness.

2. Closed trapezoidal words are Sturmian (Proposition 12). Using D’Alessandro’s structural description (Theorem 6), any non‑Sturmian trapezoidal word can be written as w = pq where p belongs to the suffix closure of ˜z_f* f and q belongs to the prefix closure of z_g* g, with (f, g) the minimal pathological pair. Lemma 8 establishes that both p and q are Sturmian (their fractional roots are conjugate to standard Sturmian words). Since a concatenation of Sturmian words remains Sturmian, a closed trapezoidal word cannot be non‑Sturmian, forcing it to be Sturmian.

3. Trapezoidal palindromes are closed and therefore Sturmian (Theorem 16). The authors give a new proof of the known equivalence “trapezoidal palindrome ⇔ Sturmian palindrome” by showing that a non‑Sturmian trapezoidal palindrome would lead to a contradiction with Lemma 5 (pathological factors cannot overlap) and Lemma 8 (p and q would have to be Sturmian). Hence every trapezoidal palindrome is closed, and by Proposition 12 it is Sturmian.

4. Characterization of special factors. Proposition 10 proves that for an open trapezoidal word the longest repeated prefix coincides with the longest right‑special factor, and dually the longest repeated suffix coincides with the longest left‑special factor. Lemma 11 translates these combinatorial properties into equalities among the parameters H_w, K_w, L_w, and R_w: for open words H_w = R_w and K_w = L_w; for closed words H_w = K_w and L_w = R_w. Lemma 14 further shows that in a closed trapezoidal word the longest special factor is a central word, linking the structure to well‑studied objects in Sturmian theory.

5. Richness. The paper recalls that every trapezoidal word is rich (contains |w| + 1 distinct palindromic factors), but richness does not imply trapezoidality (e.g., “aabbaa”).

The authors conclude with several open problems, such as determining tight bounds on the length of open trapezoidal words over larger alphabets, designing efficient algorithms to generate all closed trapezoidal words of a given length, and exploring applications of these combinatorial structures in digital geometry (coding of discrete straight lines) and cryptography.

Overall, the work deepens the understanding of trapezoidal words by linking them to Sturmian combinatorics through the closed/open dichotomy, providing precise characterizations of their special factors, and opening new avenues for both theoretical investigation and practical applications.