Symmetric peaks and symmetric valleys in Dyck paths

TL;DR

This paper extends the analysis of symmetric and asymmetric peaks and valleys in Dyck paths by providing multivariate generating functions, connecting these statistics to ordered rooted trees, and exploring monotonicity and unimodality conditions.

Contribution

It introduces multivariate generating functions for symmetric and asymmetric peaks and valleys in Dyck paths, linking these to tree structures and polyominoes, and generalizes previous formulas.

Findings

Derived multivariate generating functions for peaks and valleys.

Established a continued fraction expression connecting valleys to ordered rooted trees.

Enumerated Dyck paths with monotonic and unimodal height conditions.

Abstract

The notion of symmetric and asymmetric peaks in Dyck paths was introduced by Fl\'orez and Rodr\'{\i}guez, who counted the total number of such peaks over all Dyck paths of a given length. In this paper we generalize their results by giving multivariate generating functions that keep track of the number of symmetric peaks and the number of asymmetric peaks, as well as the widths of these peaks. We recover a formula of Denise and Simion as a special case of our results. We also consider the analogous but more intricate notion of symmetric valleys. We find a continued fraction expression for the generating function of Dyck paths with respect to the number of symmetric valleys and the sum of their widths, which provides an unexpected connection between symmetric valleys and statistics on ordered rooted trees. Finally, we enumerate Dyck paths whose peak or valley heights satisfy certain monotonicity and unimodality conditions, using a common framework to recover some known results, and relating our questions to the enumeration of certain classes of column-convex polyominoes.