Water capacity of Dyck paths

Abstract

We define the notion of capacity, the ability to contain water, for Dyck paths of semi-length n. Initially the Dyck paths attain a maximum height h and by summing over h the capacity generating function for all Dyck paths is obtained. Thereafter, we obtain the average Dyck path capacity generating function and finally an asymptotic expression for this as the semi-length increases to infinity. The proofs make use of analytic techniques such as Mellin transforms, singularity analysis and formal residue calculus.