Partial Skew Dyck Paths: A Kernel Method Approach

Skew Dyck paths are a variation of Dyck paths, where additionally to steps (1, 1) and (1 , - 1) a south–west step (- 1 , - 1) is also allowed, provided that the path does not intersect itself. Replacing the south–west step by a red south–east step, we end up with decorated Dyck paths. We analyze partial versions of them where the path ends on a fixed level j, not necessarily at level 0. We exclusively use generating functions and derive them with the celebrated kernel method.