A characterization of the words occurring as factors in a minimum number of words (Q818148)

Let \(w\) be a word of length \(k\geq 2\) on a finite alphabet \(A\) with at least two letters. Define, for \(n>k\), \({\mathcal L}(n,w)\) as the language of all words of length \(n\) on \(A\) that contain a factor equal to \(w\). The author proves that the word \(w\) for which the cardinality of \({\mathcal L}(n,w)\) is minimal are exactly the words \(w\) that are the \(k\)-th power of a single letter. Note that Reference [1] could be replaced by its published version (see the corresponding chapter in Vol. 1 of [G. Rozenberg and A. Salomaa (eds.), Handbook of formal languages. Vol. 1-3. Berlin: Springer (1997; Zbl 0866.68057)].