On the maximum number of distinct factors of a binary string

@article{Shallit1993OnTM,
title={On the maximum number of distinct factors of a binary string},
author={Jeffrey Shallit},
journal={Graphs and Combinatorics},
year={1993},
volume={9},
pages={197-200}
}

In this note we prove that a binary string of length n can have no more than 2 k+1 ?1+ ? n?k+1 2 distinct factors, where k is the unique integer such that 2 k +k ?1 n < 2 k+1 +k. Furthermore, we show that for each n, this bound is actually achieved. The proof uses properties of the de Bruijn graph. 1 I. Introduction. Let w be a string of 0's and 1's, i.e. w 2 (0 + 1). We say that z 2 (0 + 1) is a factor of w if there exist x; y 2 (0 + 1) such that