Generalizations of The Chung-Feller Theorem II

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length n with m flaws is the n-th Catalan number and independent on m. L. Shapiro [9] found the Chung-Feller properties for the Motzkin paths. Mohanty’s book [5] devotes an entire section to exploring Chung-Feller theorem. Many Chung-Feller theorems are consequences of the results in [5]. In this paper, we consider the (n,m)-lattice paths. We study two parameters for an (n,m)-lattice path: the non-positive length and the rightmost minimum length. We obtain the Chung-Feller theorems of the (n,m)-lattice path on these two parameters by bijection methods. We are more interested in the pointed (n,m)-lattice paths. We investigate two parameters for an pointed (n,m)-lattice path: the pointed non-positive length and the pointed rightmost minimum length. We generalize the results in [5]. Using the main results in this paper, we may find the Chung-Feller theorems of many different lattice paths.