The Distribution of Heights of Discrete Excursions

We compute the limiting distribution of height of a random discrete excursion with step sets consisting of one positive step 1 and arbitrary finite set of non-positive integers. The limit law is the supremum of a Brownian excursion. This is well-known for Dyck and Motzkin paths. We apply a representation of the length and height generating function in terms of certain Schur polynomials put forward in a 2008 paper by Bousquet-Melout which leads to a form of the moment generating functions amenable to a Mellin transform analysis.