# Counting Ribbon graphs

Let $G$ be a ribbon graph (sometimes called fat graph) with $v$ vertices and $e$ edges. Furthermore each vertex is of degree $d$.

Q) What is the number of $G$ with the above properties? I mean does there exist a closed formula for the number of such $G$ in terms of $v$, $e$, and $d$.

I don't know any reference or result in this area. I searched but I could not find anything of this type. Any reference, link, suggestion will be helpful.

• @Ryan You are right. I want a closed formula in terms of $v$, $e$ and $d$. I don't know anything about the number. I am editing the question. Thanks for the suggestion. – Cusp Nov 24 '14 at 6:19
• @Cusp: The data is redundant since $e = vd/2$. If you want isomorphism classes, this seems impossible (in the moral, not the technical sense) for the same reason that it is impossible to write down a closed formula for the number of isomorphism classes of degree $d$ graphs with $v$ vertices. – Andy Putman Nov 24 '14 at 6:34
A paper of I. P. Goulden and D. M. Jackson [Adv. Math. 219 (2008), no. 3, 932–951; MR2442057] gives a recursion for the total mass of dessins with ramification invariants $(3^{2m},2^{3m},\lambda)$ with $\lambda$ varying over all partitions with $m+2−2g$ parts. In the case of genus $g=0$, this recursion can be solved giving a total mass of $$\frac{F(m,0)}{6m}=\frac{4m(3m−2)!!}{m(m+2)!(m−2)!!}.$$