Moments of Dyck paths
Dyck paths of semilength \(n\) are paths from \((0,0)\) to \((2n, 0)\) with steps (1, 1) and \((1,-1)\) which lie on or above the \(x\)-axis. Strict Dyck paths have only their endpoints on the \(x\)-axis. The area under a Dyck path is the area between the Dyck path and the \(x\)-axis. \textit{D. Merlini}, \textit{R. Sprugnoli}, and \textit{M. C. Verri} [Lect. Notes Comput. Sci. 1059, 59-71 (1996)] determined the average area under Dyck paths of semilength \(n\), and \textit{L. W. Shapiro}, \textit{W.-J. Woan}, and \textit{S. Getu} [SIAM J. Algebraic Discrete Methods 4, 459-466 (1983; Zbl 0524.05006)] determined the average area under strict Dyck paths of semilength \(n\). The present paper gives a way, using generating functions, to compute higher moments of these distributions.
scientific article; zbMATH DE number 1331467