Partial skew Dyck paths -- a kernel method approach

Skew Dyck 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 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. In the second part of the paper, a dual version is studied, where the paths are read from right to left. In this way, we have two types of up-steps, not two types of down-steps, as before. A last section deals with the variation that the negative territory (below the $x$-axis) is also allowed. Surprisingly, this is more involved in terms of computations.