Enumeration of higher alternating-boundary hypermaps and extension to higher genus

Determine the general enumeration formulas for k-alternating plane hypermaps for every integer k at least 3, and extend the combinatorial enumeration framework to hypermaps on surfaces of higher genus in order to obtain combinatorial interpretations of the algebraic results supplied by topological recursion.

Background

The present paper derives formulas for accessibly pointed hypermaps for arbitrary k, but it establishes explicit formulas for the generating functions of ordinary and strongly connected hypermaps only in the cases k equals 1 and k equals 2. The general k-alternating case is deferred to a forthcoming article.

The conclusion also identifies extension to higher genus as unresolved. Such an extension is intended to provide new combinatorial interpretations of algebraic results arising from topological recursion, which currently go beyond the plane setting treated in the paper.