Partial Skew Motzkin Paths

Motzkin paths consist of up-steps, down-steps, level-steps, and never go below the $x$-axis. They return to the $x$-axis at the end. The concept of skew Dyck path \cite{Deutsch-italy} is transferred to skew Motzkin paths, namely, a left step $(-1,-1)$ is additionally allowed, but the path is not allowed to intersect itself. The enumeration of these combinatorial objects was known \cite{Qing}; here, using the kernel method, we extend the results by allowing them to end at a prescribed level $j$. The approach is completely based on generating functions. Asymptotics of the total number of objects as well as the average height are also given.