Enumeration of area-weighted Dyck paths with restricted height

Dyck paths are directed walks on Z starting at (0, 0) and ending on the line y = 0, which have no vertices with negative y-coordinates, and which have steps in the (1, 1) and (1,−1) directions [11]. We impose the additional geometrical constraint that the paths have height at most h, that is, they lie between lines y = 0 and y = h. Given a Dyck path π, we define the length n(π) to be half the number of its steps, and the area m(π) to be the sum of the starting heights of all steps in the (1, 1) direction in the path. Actually, m(π) is the value of π in the classical lattice of Dyck paths [4, 9], and is equivalent to the number of diamond plaquettes under the Dyck path. An alternative definition of the area is the sum of the heights of all steps in the Dyck path, which evaluates to n(π) + 2m(π) and is equivalent to the number of triangular plaquettes under the Dyck path. The definition used here has the advantage of enabling a more elegant and concise mathematical formulation of our results.