The enumeration of rooted cubic c-nets (Q2386823)

A simple, 3-connected map with more than 3 edges is called a c-net and \(\mathcal M_C\) denotes the collection of all rooted, cubic c-nets. Define the generating function \(H_{\mathcal M_C}(y)=\sum_{M\in\mathcal M_C}y^{n(M)},\) where \(n(M)\) denotes the number of edges in \(M\). The main result of this paper is: \[ H_{\mathcal M_C}(y)=\sum_{k\geq2}\frac{2(4k-3)!}{k!(3k-1)!}y^{3k}. \]