Explicit formulas for a family of hypermaps beyond the one-face case

TL;DR

This paper derives explicit formulas and generating functions for a family of two-face hypermaps with fixed edge-type, extending enumeration results beyond the well-studied one-face case.

Contribution

It provides the first systematic formulas for enumerating two-face hypermaps with fixed edge-type, advancing understanding beyond the one-face hypermap enumeration.

Findings

Derived explicit generating polynomials for two-face hypermaps.

Identified properties of these hypermaps related to edge-type.

Extended enumeration techniques to more complex hypermap families.

Abstract

Enumeration of hypermaps is widely studied in many fields. In particular, enumerating hypermaps with a fixed edge-type according to the number of faces and genus is one topic of great interest. However, it is challenging and explicit results mainly exist for hypermaps having one face, especially for the edge-type corresponding to maps. The first systematic study of one-face hypermaps with any fixed edge-type is the work of Jackson (Trans.~Amer.~Math.~Soc.~299, 785--801, 1987) using group characters. In 2011, Stanley obtained the generating polynomial of one-face hypermaps of any fixed edge-type expressed in terms of the backward shift operator. There are also enormous amount of work on enumerating one-face hypermaps of specific edge-types. The enumeration of hypermaps with more faces is generally much harder. In this paper, we make some progress in that regard, and obtain the generating polynomials and properties for a family of typical two-face hypermaps with almost any fixed edge-type.