Matrix integrals and generating functions for enumerating rooted hypermaps by vertices, edges and faces for a given number of darts

TL;DR

This paper introduces a recursive approach using matrix integrals to generate functions that count rooted hypermaps by vertices, edges, faces, and darts, validated through enumeration up to 13 darts.

Contribution

It presents a novel recursive method leveraging matrix integrals for enumerating rooted hypermaps, extending computational capabilities for hypermap enumeration.

Findings

Successfully enumerated all rooted hypermaps with up to 13 darts

Developed a recursive formula based on matrix integrals

Connected hypermap enumeration with bipartite quantum system analysis

Abstract

A recursive method is given for finding generating functions which enumerate rooted hypermaps by number of vertices, edges and faces for any given number of darts. It makes use of matrix-integral expressions arising from the study of bipartite quantum systems. Direct evaluation of these generating functions is then demonstrated through the enumeration of all rooted hypermaps with up to 13 darts.