On the asymptotics of the number of binary words with a given length of a maximal series Original Research Article

Let Bs(n) denote the set of binary words of length n whose longest series have the length s. The problem is to find the asymptotic expressions for the size of Bs(n) for any n and s, 1⩽s⩽n, as n→∞. Here the answer is given, whenever s⩾12 log n+2 log log n as n→∞. The remaining values of s will be considered in subsequent papers.