Enumerating symmetric pyramids in Motzkin paths

Abstract

A path in the first quadrant of the xy-plane that starts at the origin having North-East steps (X), Horizontal steps (Z), South-East steps (Y), and that ends on the x-axis is called Motzkin. A maximal pyramid is a subpath of the form XhZmYh that cannot be extended to Xh + 1ZmYh + 1. It is symmetric if it cannot be extended to any of these subpaths: Xh + 1ZmYh or XhZmYh + 1. We use generating functions to enumerate symmetric pyramids and give the asymptotic behavior of the number of symmetric pyramids. Additionally, we give combinatorial arguments to count some of the mentioned aspects.