In this paper, we develop a method to find Chung-Feller extensions for three kinds of different rooted lattice paths and prove Chung-Feller theorems for such lattice paths. In particular, we compute a generating function S(z) of a sequence formed by rooted lattice paths. We give combinatorial interpretations to the function of Chung-Feller type S(z)−yS(yz) 1−y for the generating function S(z). ...

Deng-Jye Yang, Jau-Tien Lin, Yi-Chen Chen, Shih-Chuan Liu, Fung-Jou Lu, Tien-Jye Chang, Meilin Wang, Hui-Wen Lin*, Yuan-Yen Chang* School of Health Diet and Industry Management, Chung Shan Medical University, 110, Section 1, Jianguo N. Road, Taichung 402, Taiwan Department of Nutrition, Chung Shan Medical University Hospital, 110, Section 1, Jianguo N. Road, Taichung 402, Taiwan Department of A...

1. Department of Chemistry, National Changhua University of Education, No. 1, Jin-De Road, Changhua 500, Taiwan. 2. School of Medicine, Chung Shan Medical University, No.110, Sec.1, Jianguo N.Rd.,Taichung City 402,Taiwan. 3. Division of Colorectal surgery Department of Surgery Chung Shan Medical University Hospital, No.110, Sec.1, Jianguo N. Rd.,Taichung City 402,Taiwan. 4. Department of Pathol...

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 generalizati...

Sometimes the edge set of an undirected graph is defined as a collection of doubleton subsets of the vertex set V . To unify the discussion, we will adopt the convention that the edge set E of a digraph G is a subset of V 2 and that G is undirected iff ∀i, j ∈ V ((i, j) ∈ E ↔ (j, i) ∈ E). Let G = (V,E) be a digraph with V = [n] = {1, . . . , n}. We can associate with G its adjacency matrix adj(...