# The sequence of open and closed prefixes of a Sturmian word

## TL;DR

This paper explores the oc-sequence of words, particularly Sturmian words, revealing their unique determination by this sequence and providing efficient algorithms for computing and reconstructing Sturmian words from it.

## Contribution

It establishes the unique characterization of Sturmian words via their oc-sequence and introduces linear-time algorithms for computing and reconstructing these words.

## Key findings

- The oc-sequence uniquely determines Sturmian words.
- Finite Sturmian words form a maximal class in binary factorial languages.
- Linear-time algorithms for oc-sequence computation and reconstruction.

## 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 {\it 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.