On the maximal sum of exponents of runs in a string

TL;DR

This paper establishes new upper and lower bounds on the sum of exponents of runs in strings, improving previous bounds and disproving a longstanding conjecture about their maximum sum.

Contribution

It provides improved upper bounds and a new lower bound on the sum of exponents of runs, challenging existing conjectures in string combinatorics.

Findings

Upper bound of 4.1n on sum of exponents, better than previous 5.6n

Lower bound of 2.035n, contradicting the 2n conjecture

Disproof of the conjecture that the sum is always less than 2n

Abstract

A run is an inclusion maximal occurrence in a string (as a subinterval) of a repetition $v$ with a period $p$ such that $2p \le |v|$. The exponent of a run is defined as $|v|/p$ and is $\ge 2$. 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 & Ilie (2008). The lower bound of $2.035n$, obtained using a family of binary words, contradicts the conjecture of Kolpakov & Kucherov (1999) that the maximal sum of exponents of runs in a string of length $n$ is smaller than $2n$