Equivalence classes of dessins d'enfants with two vertices

TL;DR

This paper classifies and counts dessins d'enfants with two vertices based on their edges, faces, and automorphism groups, using permutation group techniques.

Contribution

It provides explicit enumeration formulas for dessins with specific parameters and automorphism group structures, extending previous classification results.

Findings

Enumerates dessins with N edges, L faces, two vertices, and cyclic automorphism groups.

Provides enumeration for dessins with h faces of degree 2 and two vertices.

Establishes a correspondence between dessins and permutation pairs generating transitive subgroups.

Abstract

Let $N$ be a positive integer. For any positive integer $L\leq N$ 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 $h\leq N$, 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 $N$.