Generalizations of Chung-feller Theorems
نویسندگان
چکیده
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). Using our method, we first prove Chung-Feller theorems of up-down type for three kinds of rooted lattice paths. Our results are generalizations of the classical Chung-Feller theorem of up-down type for Dyck paths. We also find Motzkin paths have Chung-Feller properties of up-down type. Then we prove ChungFeller theorems of left-right type for two among three kinds of rooted lattice paths. Chung-Feller theorem of left-right type for Motzkin paths is a special case of our theorems. We also show that Dyck paths have Chung-Feller phenomenons of left-right type. By the main theorems in this paper, many new Chung-Feller theorems for rooted lattice paths are derived.
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