Applications Of Generating Function In Counting Of Lattice Paths

Counting paths and computing the area of paths play an important role in combinatorial mathematics, which can be solved by generating function. By separating a path, we can obtain a proper recurrence and then find its generating function.Making use of generating function, we can count the paths and the total area of the regions lying below the paths and above the x-axis. We deduce the generating function for some sequences. A recurrence in Dyck paths, pyramid Dyck paths, Motzkin paths and Schroder paths and their total area can be obtained by separating the path, which give us a bijection to the grammar, and then a counting sequence can be derived by the proper generating function.We add to the types of lattice paths restricted condition for deeply studying the Motzkin paths and Schroder paths. It require all concave path on the x-axis. So We define two kinds of paths which are plateau Motzkin paths and plateau Schroder paths. We get the recurrences of the sequences counting these paths by separating the path. At last, for colored plateau paths, the counting problem is also solved.