Descents in the Grand Dyck Paths and the Chung-Feller Property.

A Grand Dyck path of semilength n with m flaws is a path in the integer lattice which starts at the origin and consists of n up steps U = (1, 1) and n down steps D = (1, -1), and that has exactly m up steps below the line y = 0. The classical Chung-Feller theorem asserts that the number of grand Dyck paths of semilength n with m flaws is the nth Catalan number and is independent of m. In this paper, by using a bijection and generating functions, we prove a refinement of the Chung-Feller theorem: the number of Grand Dyck paths of semilength n having m flaws and k descents is the Narayana number N-n,N-k, and is independent of m. We also enumerate the Grand Dyck paths ending with a down step or an up step, and obtain some interesting results related to the Narayana numbers or Catalan numbers.