Generalized Chung-Feller Theorems for Lattice Paths

Generalized Chung-Feller Theorems for Lattice Paths A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University, Waltham, Massachusetts by Aminul Huq In this thesis we develop generalized versions of the Chung-Feller theorem for lattice paths constrained in the half plane. The beautiful cycle method which was developed by Devoretzky and Motzkin as a means to prove the ballot problem is modified and applied to generalize the classical Chung-Feller theorem. We use Lagrange inversion to derive the generalized formulas. For the generating function proof we study various ways of decomposing lattice paths. We also show some results related to equidistribution properties in terms of Narayana and Catalan generating functions. We then develop generalized Chung-Feller theorems for Motzkin and Schröder paths. Finally we study generalized paths and the analogue of the Chung-Feller theorem for them.

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