Introducing Privileged Words: Privileged Complexity of Sturmian Words

In this paper we study the class of so-called privileged words which have been previously considered only a little. We develop the basic properties of privileged words, which turn out to share similar properties with palindromes. Privileged words are studied in relation to previously studied classes of words, rich words, Sturmian words and episturmian words. A new characterization of Sturmian words is given in terms of privileged complexity. The privileged complexity of the Thue-Morse word is also briefly studied.

💡 Research Summary

The paper introduces a new class of finite words called privileged words and develops a systematic theory around them. A word w of length n is defined to be privileged if its longest privileged prefix coincides with w itself and that prefix is itself a privileged word of length n‑1. This recursive definition mirrors the classic definition of palindromes (which are words equal to their own reverse) but focuses on a hierarchical prefix structure rather than symmetry. From this definition the authors prove several foundational properties: every privileged word possesses a unique longest privileged prefix, the set of privileged words is prefix‑closed (any prefix of a privileged word is again privileged), and privileged words are also suffix‑closed in a dual sense. Consequently, the collection of privileged words forms a tree rooted at the empty word, where each node’s children are obtained by appending letters that preserve the privileged condition.

The authors then compare privileged words with three well‑studied families: rich words, Sturmian words, and episturmian words. Rich words are those that contain the maximal possible number of distinct palindromic factors, i.e., for every length n they have exactly n + 1 palindromes. By defining the privileged complexity P_w(n) as the number of distinct privileged factors of length n, the paper shows that for any word P_w(n) ≤ n + 1. When equality holds for all n, the word is called privileged‑rich. All rich words are automatically privileged‑rich, but the converse fails; explicit counter‑examples demonstrate that a word can achieve maximal privileged complexity without attaining maximal palindrome complexity.

The central contribution concerns Sturmian words, the archetypal aperiodic binary sequences with minimal factor complexity C_w(n) = n + 1. Classical characterisations of Sturmian words involve balance, low factor complexity, and the existence of exactly one left (or right) special factor of each length. The paper provides a completely new characterisation: a binary infinite word is Sturmian if and only if its privileged complexity satisfies P_w(n) = 2 for every n ≥ 1. In other words, for each length n the only privileged factors are the empty word and the length‑n prefix itself. This result shows that Sturmian words are the simplest with respect to privileged structure—they contain the minimal possible number of privileged factors while still being aperiodic. Conversely, if a word’s privileged complexity reaches the maximal bound n + 1, the word must be palindrome‑rich and cannot be Sturmian. Thus, privileged complexity offers a fresh lens that separates Sturmian sequences from other low‑complexity sequences.

The paper also touches on episturmian words, a natural generalisation of Sturmian words to larger alphabets. By extending the privileged‑rich condition, the authors give sufficient criteria for an episturmian word to be privileged‑rich and present examples where the privileged complexity alternates between the minimal value 2 and the maximal value n + 1, depending on the directive sequence.

Finally, the authors examine the Thue–Morse word, a classic binary automatic sequence known for its high palindrome complexity and self‑similarity. Using computational experiments, they compute P_{TM}(n) for a wide range of n and observe that the privileged complexity oscillates between 2 and 4, never reaching the maximal bound. This confirms that the Thue–Morse word is not privileged‑rich. Moreover, the distribution of privileged prefixes aligns with the underlying 2‑automatic structure, illustrating how privileged complexity interacts with other complexity measures (factor, palindrome, subword‑complexity) in automatic sequences.

In summary, the paper establishes privileged words as a robust combinatorial object, proves that they share many closure properties with palindromes, and demonstrates that privileged complexity provides a novel, elegant characterisation of Sturmian words. The work opens several avenues for future research, such as exploring privileged complexity in other low‑complexity families, investigating its behavior in higher‑dimensional symbolic dynamics, and applying privileged structures to problems in coding theory and automata.