Equivalence classes of dessins d'enfants with two vertices
Madoka Horie

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 be a positive integer. For any positive integer and any positive divisor of , we enumerate the equivalence classes of dessins d'enfants with edges, faces and two vertices whose automorphism groups are cyclic of order . Further, for any non-negative integer , we enumerate the equivalence classes of dessins with edges, faces of degree with , and two vertices, whose automorphism groups are cyclic of order . Our arguments are essentially based upon a natural one-to-one correspondence of the equivalence classes of all dessins with edges to the equivalence classes of all pairs of permutations with components generating transitive subgroups of the symmetric group of degree .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsFinite Group Theory Research · graph theory and CDMA systems · Coding theory and cryptography
