## Some enumerations on non-decreasing Dyck paths

- Nov. 13, 2020
- 2:30 p.m.

## Abstract

A *Dyck path* is a lattice path in the first quadrant of the $xy$-plane that starts at the origin and ends on the $x$-axis. It consists of the same number of North-East (U) and South-East (D) steps. A *pyramid* is a subpath of the form $U^nD^n$. A *valley* is a subpath of the form $DU$. The height of a valley is the $y$-coordinate of its lowest point. A Dyck path is called *non-decreasing* if the heights of its valleys form a non-decreasing sequence from left to right.

In this talk, we count several aspects of non-decreasing Dyck paths. We count, for example, the number and weight of pyramids and numbers of primitive paths. In the end of the talk, we introduce the concept of symmetric pyramids and count them. Throughout the talk, we give connections (bijective relations) between non-decreasing Dyck paths with other objects of the combinatorics. Some examples are, words, trees, polyominoes. This is a joint work with Éva Czabarka, José L. Ramírez, and Leandro Junes.