The Number of Rooted Maps with a Fixed Number of Vertices

Let \(T_{g}(m,n)\) (respectively, \(P_{g}(m, n)\)) be the number of rooted maps, on an orientable (respectively, non-orientable) surface of type \(g\), which have \(m\) vertices and \(n\) faces. Bender, Canfield and Richmond [3] obtained asymptotic formulas for \(T_{g}(m,n)\) and \(P_{g}(m,n)\) when \(\epsilon \leq m/n \leq 1/\epsilon\) and \(m,n \to \infty\). Their formulas cannot be extended to the extreme case when \(m\) or \(n\) is fixed. In this paper, we shall derive asymptotic formulas for \(T_{g}(m,n)\) and \(P_{g}(m,n)\) when \(m\) is fixed and derive the distribution for the root face valency. We also show that their generating functions are algebraic functions of a certain form. By the duality, the above results also hold for maps with a fixed number of faces.