On the average number of regularities in a word

We study the average number of powers and runs occurring in a word of length n drawn from an alphabet of size σ  . We show that a word contains nσ(r−1)−1+o(n) powers of exponent r  , at most nσ+o(n) runs, and also (1+2σ−1)n+o(n) palindromes. We also explore their abelian variants and prove that a binary word contains Θ(n32) abelian squares on the average, while its number of abelian cubes is related to Franel numbers. Finally, as a consequence, we show that a binary word has almost surely O(n32) abelian squares.