An asymptotic lower bound for the maximal-number-of-runs function

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. © 2006 Czech Technical University in Prague, Czech Republic.