An asymptotic Lower Bound for the Maximal Number of Runs in a String

@article{Franek2008AnAL,
  title={An asymptotic Lower Bound for the Maximal Number of Runs in a String},
  author={Frantisek Franek and Qian Yang},
  journal={Int. J. Found. Comput. Sci.},
  year={2008},
  volume={19},
  pages={195-203}
}
An asymptotic lower bound for the maxrun function ρ(n) = max {number of runs in string x | all strings x of length n} is presented. More precisely, it is shown that for any ε > 0, (α−ε)n is an asymptotic lower bound, where α = 3 1+ √ 5 ≈ 0.927. A recent construction of an increasing sequence of binary strings “rich in runs” is modified and extended to prove the result. 

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