Dyck Paths With No Peaks At Height k

A Dyck path of length 2n is a path in two-space from (0, 0) to (2n, 0) which uses only steps (1, 1) (north-east) and (1,−1) (south-east). Further, a Dyck path does not go below the x-axis. A peak on a Dyck path is a node that is immediately preceded by a north-east step and immediately followed by a south-east step. A peak is at height k if its y-coordinate is k. Let Gk(x) be the generating function for the number of Dyck paths of length 2n with no peaks at height k with k ≥ 1. It is known that G1(x) is the generating function for the Fine numbers (sequence A000957 in [6]). In this paper, we derive the recurrence Gk(x) = 1 1− xGk−1(x) , k ≥ 2, G1(x) = 2 1 + 2x+ √ 1− 4x . It is interesting to see that in the case k = 2 we get G2(x) = 1+xC(x), where C(x) is the generating function for the ubiquitous Catalan numbers (A000108). This means that the number of Dyck paths of length 2n + 2, n ≥ 0, with no peaks at height 2 is the Catalan number cn = 1 n+1 ( 2n n ) . We also provide a combinatorial proof for this last fact by introducing a bijection between the set of all Dyck paths of length 2n+ 2 with no peaks at height 2 and the set of all Dyck paths of length 2n.