Equivalence classes of dessins d'enfants with two vertices

Let $N$ be a positive integer. For any positive integer $L\le N$ and any positive divisor $r$ of $N$, we enumerate the equivalence classes of dessins d’enfants with $N$ edges, $L$ faces and two vertices whose automorphism groups are cyclic of order $r$. Further, for any non-negative integer $h$, we enumerate the equivalence classes of dessins with $N$ edges, $h$ faces of degree $2$ with $h\le N$, and two vertices, whose automorphism groups are cyclic of order $r$. Our arguments are essentially based upon a natural one-to-one correspondence of the equivalence classes of all dessins with $N$ edges to the equivalence classes of all pairs of permutations with components generating transitive subgroups of the symmetric group of degree $N$.