The number of labeled tetracyclic series-parallel blocks

A series-parallel graph is a graph that does not contain a complete graph with four vertices as a minor. Such graphs are used in the construction of reliable communication networks. Let $TB(n)$ be the number of labeled series-parallel tetracyclic blocks with $n$ vertices. The formula $TB(n)=\dfrac{n!}{80640}(n^5+30n^4+257n^3+768n^2+960n+504)\dbinom{n-3}{3}$ is obtained. It is proved that with a uniform probability distribution, the probability that the labeled tetracyclic block is a series-parallel graph is asymptotically $3/11$.