Affiliations: The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, P.O. Wits 2050, Johannesburg, South Africa, Charlotte.Brennan@wits.ac.za | Department of Mathematics, University of Venda, Private bag x5050, Thohoyandou, 0950, South Africa, Simon.Mavhungu@univen.ac.za
Note: [] Address for correspondence: University of the Witwatersrand, P.O. Wits 2050, Johannesburg, South Africa
Abstract: A Dyck path is a non-negative lattice path in $\mathbb{N}^2$ starting at the origin, where only two types of steps are allowed: the diagonal up step (1, 1) and the diagonal down step (1, −1). The length of the path is the total number of unit steps. We consider paths of length n, ending at the point (n, i). A path is considered to be closed if i = 0 and open if i ≥ 0. This aim of this paper is to find asymptotic expressions for the first and second moments of the number of visits to a certain level r a Dyck path makes. We investigate open and closed paths separately where we investigate the two different cases r > 0 and r = 0. We use generating functions which are found using recursions. These recursions are solved using matrix algebra. Asymptotic expressions for the expected value and variance are obtained using singularity analysis where the generating functions are expanded around the dominant singularities.
