Asymptotics of the Average Height of $2$–Watermelons with a Wall

Abstract

We generalize the classical work of de Bruijn, Knuth and Rice (giving the asymptotics of the average height of Dyck paths of length $n$) to the case of $p$–watermelons with a wall (i.e., to a certain family of $p$ nonintersecting Dyck paths; simple Dyck paths being the special case $p=1$.) An exact enumeration formula for the average height is easily obtained by standard methods and well–known results. However, straightforwardly computing the asymptotics turns out to be quite complicated. Therefore, we work out the details only for the simple case $p=2$.