Abstract :
The study of the combinatorial properties of strings of symbols from a finite alphabet, also referred to as words, is profoundly connected to numerous fields such as biology, computer science, mathematics, and physics. Partial words have long been known. They have been around in information and coding theory as well as molecular biology. While a word can be described by a total function, a partial word can be described by a partial function. More precisely, a partial word of length n over a finite alphabet A is a partial function from {0, … , n − 1} into A. Elements of {…, n − 1 without an image are called holes. A word is just a partial word without holes. The notion of period of a word is central in combinatorics on words. In the case of partial words, there are two notions: one is that of period, and the other is that of local period. There are many fundamental results on periods of words. Among them is the well-known periodicity result of Fine and Wilf which intuitively determines how far two periodic events have to match in order to guarantee a common period. This result was extended to partial words with one hole by Berstel and Boasson. We can define the set PERo of all words of maximal length for which Fine and Wilfʹs result does not apply. Similarly, we can define the set PER1 of all partial words with one hole of maximal length for which Berstel and Boassonʹs result does not apply. It is well known that PERo contains a unique word (up to a renaming) that is binary. In this note, we extend this result to PER1.
