The Riordan Group and Symmetric Lattice Paths

Details The Riordan Group and Symmetric Lattice Paths The Riordan Group and Symmetric Lattice Paths

2009-06-10

arxiv.org/abs/0906.1844v1

Mathematics

Combinatorics

16 pages

Scientific paper

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 $2n.$

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