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 S_n be the set of all skew Motzkin paths of length n and let s_n = |S_n|. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence {s_n}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.