On the maximal sum of exponents of runs in a string

A run is an inclusion maximal occurrence in a string (as a subinterval) of a repetition v with a period p   such that 2p⩽|v|2p⩽|v|. The exponent of a run is defined as |v|/p|v|/p and is greater than or equal to 2. We show new bounds on the maximal sum of exponents of runs in a string of length n. Our upper bound of 4.1n is better than the best previously known proven bound of 5.6n by Crochemore and Ilie (2008). The lower bound of 2.035n, obtained using a family of binary words, contradicts the conjecture of Kolpakov and Kucherov (1999), that the maximal sum of exponents of runs in a string of length n is smaller than 2n.