On the average depth of a prefix of the Dycklanguage D1

If D₁ is the Dycklanguage with one type of bracket then the level of a bracket in a word w ∈ D₁ 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 D₁ and show that the average depth is given for all ε>0 by

$\text{(ln2)}\text{2πn}\text{?}\text{3}\text{2}\text{+}\text{1}\text{8}\text{2π}\text{n}\text{[4ln2?(?1)}{}^{\mathrm{n}}\text{(2ln2?1)]+O(}\text{ln n}\text{n}{}^{\mathrm{1.5?\; ?}}$

. The variance is asymptotically for all ?>0

$\text{(}\text{π}\text{3}\text{?2 ln}{}^2\text{2)πn+}\text{5π}{}^2\text{18}\text{?2πln}{}^2\text{2+}\text{1}\text{12}\text{?π(?1)}{}^{\mathrm{n}}\text{(}\text{π}\text{24}\text{? ln}{}^2\text{2+}\text{1}\text{2}\text{ln2)+O(}\text{ln n}\text{n}{}^{\mathrm{1??}}$

. Furthermore, we derive several enumeration results describing the distribution of the number of certain prefixes of length n.