Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises

Source abstract

Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in [Formula: see text] starting at the origin made of steps [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], where two down-steps cannot be consecutive. We enumerate MAP and their prefixes avoiding peaks (respectively, valleys, respectively, double rise) according to the length, the type of the last step, and the height of its end-point. We express our results using Riordan arrays. Finally, we provide constructive bijections between these paths and restricted Dyck and Motzkin paths.