Average number of occurrences of repetitions in a necklace

In this paper, we consider the average number of occurrences of primitively rooted repetitions in a necklace. First, we define circular repetitions for a string and show the average number of them. Using these results, we can obtain the average number of squares, cubes, runs and cubic runs and the average sum of exponents of runs and cubic runs in a necklace, exactly.