The sequence of open and closed prefixes of a Sturmian word

Abstract

A finite word is closed if it contains a factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We are interested in the oc-sequence of a word, which is the binary sequence whose n-th element is 0 if the prefix of length n of the word is open, or 1 if it is closed. We exhibit results showing that this sequence is deeply related to the combinatorial and periodic structure of a word. In the case of Sturmian words, we show that these are uniquely determined (up to renaming letters) by their oc-sequence. Moreover, we prove that the class of finite Sturmian words is a maximal element with this property in the class of binary factorial languages. We then discuss several aspects of Sturmian words that can be expressed through this sequence. Finally, we provide a linear-time algorithm that computes the oc-sequence of a finite word, and a linear-time algorithm that reconstructs a finite Sturmian word from its oc-sequence

Characteristic Parameters and Special Trapezoidal Words

“…A few related problems remain open. In particular, in [5] the oc-sequence for (prefixes of) characteristic Sturmian words was characterized, see Theorem 11. The general trapezoidal case, and even the non-standard Sturmian one, is still open.…”

“…In particular, the case of (prefixes of) the Fibonacci word was completely characterized, and this was later (cf. [5]) extended to all characteristic Sturmian words. Our first aim, explored in Section 3, is to extend some of those arguments to the general trapezoidal case, with respect to the values of characteristic parameters for closed and open prefixes.…”

Closed Ziv-Lempel factorization of the $m$-bonacci words

“…The shortest (resp., longest) closed factorization of a string is obtained by factorizing it into shortest (resp., longest) closed factors. In [7], A. De Luca et al studied closed prefixes of Sturmian words and introduced the oc-sequence of a word w, as oc(w), which is the binary sequence whose n-th term is 1 if the length-n prefix of w is closed, or 0 if it is open.…”

“…The notion of the oc-sequence of a word w is introduced in [7]. It is a binary sequence whose n-th element is 1 if the length-n prefix of w is closed; otherwise, it is 0.…”

Minimal Forbidden Factors of Circular Words

“…But it is known that the longest repeated prefix of u n xyu n is u n (cf. [14]), so u n x cannot appear in u n xyu n .…”