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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, andsymmetric Schr\"oder paths of length $2n$, respectively. By using Riordan groupmethods we obtain six identities relating $d_{n}$, $m_{n}$, and $s_{n}$ andalso give two of them combinatorial proofs. Finally, we investigate somerelations satisfied by the generic element of some special Riordan arrays andget the average mid-height and the average number of points on the x-axis ofsymmetric Dyck paths of length $2n.$

The Riordan Group and Symmetric Lattice Paths