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28 March 2024
 
  » arxiv » 0903.0705

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Generalizations of Chung-Feller Theorem II
Jun Ma ; Yeong-nan Yeh ;
Date 4 Mar 2009
AbstractThe 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.
Source arXiv, 0903.0705
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