The maximal number of cubic runs in a word

Abstract : A run is an inclusion maximal occurrence in a word (as a subinterval) of a factor in which the period repeats at least twice. The maximal number of runs in a word of length n has been thoroughly studied, and is known to be between 0.944n and 1.029n. The proofs are very technical. In this paper we investigate cubic runs, in which the period repeats at least three times. We show the upper bound on their maximal number, cubic-runs(n), in a word of length n: cubic-runs(n) < 0.5n. The proof of linearity of cubic-runs(n) utilizes only simple properties of Lyndon words and is considerably simpler than the corresponding proof for general runs. For binary words, we provide a better upper bound cubic-runs2(n) < 0.48n which requires computer-assisted verification of a large number of cases. We also construct an infinite sequence of words over a binary alphabet for which the lower bound is 0.41n.