New Lower Bounds for the Maximum Number of Runs in a String

TL;DR

This paper establishes a new, improved asymptotic lower bound for the maximum number of runs in a string, simplifying previous proofs and constructions.

Contribution

It introduces a tighter lower bound of approximately 0.944542 for the maximum runs, surpassing prior bounds, with a simpler proof and construction.

Findings

New lower bound of 0.944542 for maximum runs

Simpler proof and construction compared to previous work

Improves upon the previous bound of 0.927

Abstract

We show a new lower bound for the maximum number of runs in a string. We prove that for any e > 0, (a -- e)n is an asymptotic lower bound, where a = 56733/60064 = 0.944542. It is superior to the previous bound 0.927 given by Franek et al. Moreover, our construction of the strings and the proof is much simpler than theirs.