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

Enumeration of hypermaps (or Grothendieck's dessins d'enfants) 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. The first systematic study of hypermaps with one face and any fixed edge-type is the work of Jackson (1987) [23] using group characters. Stanley later (2011) obtained the genus distribution polynomial of one-face hypermaps of any fixed edge-type expressed in terms of the backward shift operator. There is also enormous amount of work on enumerating one-face hypermaps of specific edge-types. Hypermaps with more faces are generally much harder to enumerate and results are rare. Our main results here are formulas for the genus distribution polynomials for a family of typical two-face hypermaps including almost all edge-types, the purely imaginary zeros property of these polynomials, and the log-concavity of the coefficients.