The Riordan Group and Symmetric Lattice Paths

TL;DR

This paper explores symmetric lattice paths using Riordan group techniques, deriving identities and combinatorial proofs, and analyzing properties like mid-height and points on the x-axis of symmetric Dyck paths.

Contribution

It introduces new identities relating symmetric Dyck, Motzkin, and Schr"oder paths using Riordan group methods and provides combinatorial proofs for some of these identities.

Findings

Derived six identities relating symmetric lattice paths.

Provided two combinatorial proofs of these identities.

Analyzed average mid-height and points on the x-axis of symmetric Dyck paths.

Abstract

In this paper, we study symmetric lattice paths. Let $d_{n}$, $m_{n}$, and $s_{n}$ denote the number of symmetric Dyck paths, symmetric Motzkin paths, and symmetric Schr\"oder paths of length $2n$, respectively. By using Riordan group methods we obtain six identities relating $d_{n}$, $m_{n}$, and $s_{n}$ and also give two of them combinatorial proofs. Finally, we investigate some relations satisfied by the generic element of some special Riordan arrays and get the average mid-height and the average number of points on the x-axis of symmetric Dyck paths of length $2n.$