The Riordan Group and Symmetric Lattice 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 Schröder 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 $2n.$

16 pages