A Natural Extension of Catalan Numbers

A Dyck path is a lattice path in the plane integer lattice Z × Z consisting of steps (1, 1) and (1,−1), each connecting diagonal lattice points, which never passes below the x-axis. The number of all Dyck paths that start at (0, 0) and finish at (2n, 0) is also known as the nth Catalan number. In this paper we find a closed formula, depending on a non-negative integer t and on two lattice points p1 and p2, for the number of Dyck paths starting at p1, ending at p2, and touching the x-axis exactly t times. Moreover, we provide explicit expressions for the corresponding generating function and bivariate generating function.