Counting Morse functions on the 2-sphere

This paper was motivated by work of Arnold where he explains how to count "snakes", i.e. Morse functions on the real axis with prescribed behavior at infinity. This leads immediately to a count of excellent Morse functions on the circle, where following Thom's terminology, excellent means that no two critical points lie on the same level set. We explain how to count equivalence classes of excellent Morse functions on the 2-sphere. We consider two equivalence relations. The geometric relation (two functions are equivalent if one can be obtained from the other via compositions with an orientation preserving diffeomorphism of the sphere and an orientation preserving diffeomorphism of the real axis) and the homology relation (two functions are equivalent if the Betti numbers of the sublevel sets undergo similar changes when crossing a critical value).