Abstract
A Dyck path is a lattice path in the first quadrant of the xy-plane that starts at the origin and ends on the x-axis and has even length. This is composed of the same number of North-East (X) and South-East (Y) steps. A peak and a valley of a Dyck path are the subpaths XY and YX, respectively. A peak is symmetric if the valleys determining the maximal pyramid containing the peak are at the same level. In this paper we give recursive relations, generating functions, as well as closed formulas to count the total number of symmetric peaks and asymmetric peaks. We also give an asymptotic expansion for the number of symmetric peaks.
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