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arXiv · 0812.2978

Generalizations of Chung-Feller Theorem

Jun Ma·Yeong-Nan Yeh
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Abstract

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this paper, we find the connections between these two Chung-Feller theorems. We focus on the weighted versions of three classes of lattice paths and give the generalizations of the above two theorems. We prove the Chung-Feller theorems of Dyck type for these three classes of lattice paths and the Chung-Feller theorems of Motzkin type for two of these three classes. From the obtained results, we find an interesting fact that many lattice paths have the Chung-Feller properties of both Dyck type and Motzkin type.

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Jun Ma, Yeong-Nan Yeh. 2008-12-16. Generalizations of Chung-Feller Theorem. https://arxiv.org/abs/0812.2978

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