Morse Functions of the Two Sphere

We count how many ”different” Morse functions exist on the 2-sphere. There are several ways of declaring that two Morse functions f and g are ”indistinguishable” but we concentrate only two natural equivalence relations: homological (when the regular sublevel sets f and g have identical Betti numbers), and geometric (when f is obtained from g via global, orientation preserving changes of coordinates on S and R). The count of homological classes is reduced to a count of lattice paths confined to the first quadrant. The count of geometric classes is reduced to a count of certain labelled trees. We produce a two-parameter recurrence which can be encoded by a first order quasilinear pde. We solve this equation using the classical method of characteristics and we produce a closed form description of the exponential generating function of the numbers of geometric classes.