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 X^hZ^mY^h that cannot be extended to X^h + 1Z^mY^h + 1. It is symmetric if it cannot be extended to any of these subpaths: X^h + 1Z^mY^h or X^hZ^mY^h + 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.