Generalizations of Chung-Feller Theorem II

TL;DR

This paper extends the Chung-Feller theorem to various lattice paths, introducing new parameters and bijection methods to establish generalized Chung-Feller properties for pointed and unpointed paths.

Contribution

It generalizes Chung-Feller theorems to $(n,m)$-lattice paths and pointed paths, using bijections to relate new parameters to classical results.

Findings

Chung-Feller theorems established for $(n,m)$-lattice paths.

Generalization of results to pointed lattice paths.

Framework to derive Chung-Feller theorems for various lattice paths.

Abstract

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.