Improved Bounds on the Sum of Exponents of Runs in a String

A substring of a word is a run if it is at least twice as long as its minimum period and cannot be extended to either side with the same period. The exponent of a run is the quotient of its length and its minimum period. ρ(n) is the maximum number of runs in a string of length n, while σ(n) is the maximum sum of exponents of runs in a string of length n. While quite tight bounds on ρ(n) are known (0.944575712n ≤ ρ(n) ≤ n), the best upper bound on σ(n) is 3n whereas the best lower bound on σ(n) is 2.035n. In this paper, we improve the upper bound on σ(n) to 2.3n and the lower bound on σ(n) to 2.04448n. We also provide an improved upper bound on σ(n) of 2.2n in the case of a binary alphabet. Our results are achieved using a combination of theoretical and computer-based approaches.