Symmetric and asymmetric peaks or valleys in (partial) Dyck paths

TL;DR

This paper provides a bijective enumeration of symmetric and asymmetric peaks and valleys in (partial) Dyck paths, refining previous results and offering detailed combinatorial insights into these path statistics.

Contribution

It introduces bijective methods to enumerate symmetric and asymmetric peaks and valleys in (partial) Dyck paths, extending prior generating function approaches.

Findings

Enumerates symmetric and asymmetric peaks and valleys in Dyck paths.

Refines previous results by Flórez, Ramírez, and Elizalde.

Provides combinatorial bijections for path statistics.

Abstract

The concepts of symmetric and asymmetric peaks in Dyck paths were introduced by Fl\'{o}rez and Ram\'{\i}rez, who counted the total number of such peaks over all Dyck paths of a given length. Elizalde generalized their results by giving multivariate generating functions that keep track of the number of symmetric peaks and the number of asymmetric peaks. Elizalde also considered the analogous notion of symmetric valleys by a continued fraction method. In this paper, mainly by bijective methods, we devote to enumerating the statistics "symmetric peaks", "asymmetric peaks", "symmetric valleys" and "asymmetric valleys" of weight $k+1$ over all (partial) Dyck paths of a given length. Our results refine some consequences of Fl\'{o}rez and Ram\'{\i}rez, and Elizalde.