Counting maximal-exponent factors in words

The topic of repeating segments in words is one of major interest in combinatorics on words. The topic has been studied for more than a century by many authors after the seminal work [9] which described infinite words containing no consecutive occurrences of the same factor. Beyond the theoretical aspect of questions related to redundancies in words, repetitions, also called repeats in the following, are often the base for string modelling adapted to compression coding. They play an important role in run-length compression and in Ziv–Lempel compression, e.g., [4]. Moreover, repetitions receive considerable attention in connection with the analysis of genetic sequences. Their occurrences are called tandem repeats, satellites or SRS and accept some notion of approximation. The existence of some palindromic repeats is crucial for the prediction of the secondary structure of RNA molecules influencing their biological functions, see [5]. Repetitions are composed of consecutive occurrences of the same factor. Their occurrences have been extended to runs [8], maximal periodic factors, and their number has been shown to be less than the word length n [3] (see also [6]) and even further less than 22n/23 [7]. In this article we consider factors that repeat non consecutively in a given word of length n. They are of the form uvu, where u is their longest border (factor occurring both at the beginning and end of the word). Their exponent, defined as the ratio of their length over their smallest period length, that is, |uvu|/|uv|, is smaller than 2. The number of occurrences of these factors may be quadratic with respect to the word length even if they are restricted to non extensible occurrences. This is why we focus on factors having the maximal exponent among all factors occurring in a square-free word. They are called maximal-exponent factors, MEFs in short, and thus have all the same exponent. ∗This represents an extended abstract of a work accepted for publication in Theor. Comput. Sci., doi:10.1016/j.tcs.2016.02.035