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 {\it oc-sequence} of a word, which is the binary sequence whose -th element is if the prefix of length of the word is open, or 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.
Keywords
Cite
@article{arxiv.1701.01580,
title = {The sequence of open and closed prefixes of a Sturmian word},
author = {Alessandro De Luca and Gabriele Fici and Luca Q. Zamboni},
journal= {arXiv preprint arXiv:1701.01580},
year = {2018}
}
Comments
Published in Advances in Applied Mathematics. Journal version of arXiv:1306.2254