Lattice paths in R² and R³

The focus in this thesis is on lattice paths in R² and R³. Three ways of counting the number of lattice paths from the origin to a point inR² and R³ are examined; the recurrence relation, the closed form expression, and the generating function. In the first two chapters, lattice paths from (0,0) to (p, q) are discussed, first without restriction andthen with restrictions based on whether the path may or may not cross or touch the line y = X. In the first chapter, lattice paths are not allowed to move diagonally, and it is shown how the ballot problem is an application. In the second chapter, lattice paths are allowed diagonal moves. In the third and fourth chapters, lattice paths from (0, 0,0) to(p, q, r) are also discussed first without restriction and then with the restrictions that foreach point (i, j, k) on the lattice path i ≥ j, i > j, or i ≥ j ≥ k. In the third chapter, diagonal moves are not allowed, and the ballot problem is discussed. In the fourth chapter,diagonal moves are allowed.