Bannai et al. method proves the d-step conjecture for strings

Inspired by the d-step approach used for investigating the diameter of polytopes, the following d-step conjecture was introduced by Deza and Franek : the number of runs in a string of length n with d distinct symbols is at most n− d. Bannai et al. showed that the number of runs in a string is less than its length by mapping each run to a set of starting positions of Lyndon roots. We show that Bannai et al. method proves that the d-step conjecture for runs holds, and stress the structural properties of run-maximal strings. In particular, we show that, up to relabelling, there is a unique run-maximal string of length 2d with d distinct symbols. As corollary, we obtain a slight improvement of Bannai et al. bound : the number of runs in a string of length n is at most n− 4 for n ≥ 9.