On the average depth of a prefix of the Dycklanguage D1

If D, is the Dycklanguage with one type of bracket then the level of a bracket in a word w E D, is defined as the number of preceding opening brackets minus the number of preceding closing brackets. The depth of a Dyckword w is the maximum level of a bracket appearing in w. In this paper we derive an explicit expression for the average depth of a prefix of length n of the Dycklanguage D, and show that the average depth is given for all E > 0 by