Skew Motzkin paths

In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U = (1, 1), down steps D = (1,−1), horizontal steps H = (1, 0), and left steps L = (−1,−1), and such that up steps never overlap with left steps. Let Sn be the set of all skew Motzkin paths of length n and let sn = |Sn|. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence {sn}n≥0. Then we present several involutions on Sn and consider the number of their fixed points. Finally we consider the enumeration of some statistics on Sn.