On \(S-(p, q)\)-Dyck paths

Abstract

Bizley [J. Inst. Actuar. 80 (1954), 55-62] studied a generalization of Dyck paths from \((0,0)\) to \((pn, gn)\) (\(\gcd(p,q) = 1\)), which never go below the line \(py = qx\) and are made of steps in \(\{(0, 1), (1,0)\}\), called the step set, and calculated the number of such paths. In this paper, we mainly generalize Bizley’s results to an arbitrary step set \(S\). We call these paths \(S\)-\((p,q)\)-Dyck paths, and give explicit enumeration formulas for such paths. In addition, we provide a proof of these formulas using the method presented in Gessel [J. Combin. Theory Ser. A 28 (1980), no. 3, 321-337]. As applications, we calculate some examples which generalize the classical Schröder and Motzkin numbers.