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

Let B-s(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 B-s(n) for any it and s, 1 less than or equal to s less than or eq ual to n, as n --> infinity. Here the answer is given, whenever s greater t han or equal to 1/2 log n divided by 2 log log n as n --> infinity. The rem aining values of s will be considered in subsequent papers. (C) 2001 Elsevi er Science B.V. All rights reserved.