THE NUMBER OF ROOTED MAPS ON AN ORIENTABLE SURFACE

Let mg(n) be the number of rooted n edged maps on an orientable surface of genus g > 0. The generating function Mg(x) = Σ mg(n) xn is a rational function of ρ{variant} = (1 - 12x) 1 2 whose denominator factors completely into powers of ρ{variant}, ρ{variant} + 2, and ρ{variant} + 5. We calculate M2(x) and M3(x). Unfortunately, we have not been able to discern a pattern in the sequence Mg(x) from the values for g ≤ 3. © 1991.