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 S-n and consider the number of their fixed points. Finally we consider the enumeration of some statistics on S-n .