On the enumeration of labeled series-parallel $k$-cyclic $2$-connected graphs

We deduce an explicit formula for the number of labeled series-parallel $k$-cyclic $n$-vertex $2$-connected graphs and find the corresponding asymptotics for a large number of vertices and a fixed $k$. Under the uniform probability distribution, an asymptotic formula is obtained for the probability that a random $n$-vertex $k$-cyclic $2$-connected graph with a large number of vertices and a fixed $k$ is a series-parallel graph. Tab. 1, bibliogr. 11.