Generalized m-Schröder paths and the Chung-Feller property

In this paper, we introduce a generalization of m-Schröder paths. For a fixed positive integer m, the generalized m-Schröder paths are lattice paths that start at (0, 0), use the steps U = (1, 1), H = (1, 0), \mathrm{V_1} = (0,-1), and \mathrm{V_2} = (0,-2) which are weighted respectively by 1, h, a and b, remain weakly above theline y =m-1/m x, and end on this line. We use generating functions and Riordan arrays to discuss the enumeration of the partial generalized m-Schröder paths and the free generalized m-Schröder paths, and obtain a Chung-Feller property. In particular, when h = a = b = 1, we find that the number of generalized m-Schröder paths of order n equals the number of hybrid (m + 1)-ary trees with n internal nodes.