Enumeration Of Peaks In Dyck, Motzkin And Schr(?)der Paths

Dyck paths, Motzkin Paths and Schroder paths are very important combinatorial struc-tures in enumerative combinatorics. Besides their close connections with other combina-torial structures like trees, restricted permutations, orthogonal polynomials, continued frac-tions, etc., their applications also arise from fields as statistics, random processes and bioin-formatics. In this paper we mainly study the statistic "number of peaks" in these lattice paths. In [16] A. Regev compute the number of peaks (humps) in Motzkin paths, and notice that the number of generalized Motzkin paths of order n equals one plus twice the number of peaks in all Motzkin paths of order n, and ask for a bijective proof for this identity. In this paper we give such a bijection, from which we give the number of peaks in Dyck paths, Motzkin paths and Schroder paths. Using this bijection, we give a new proof that the number of Dyck paths of order n with k peaks is the Narayana number. By double counting gener-alized Schroder paths, we also find an interesting identity involving products of binomial coefficients, a slightly different form of this identity appears in [21] Page 115 ex.3(g).Another result of this paper is a bijection via RSK algorithm between Motzkin paths and Standard Young Tableaux with at most three rows.