Matrix integrals and generating functions for permutations and one-face rooted hypermaps

TL;DR

This paper derives closed-form generating functions for counting one-face rooted hypermaps with a fixed number of darts, vertices, and edges using matrix integrals related to quantum systems, and outlines methods for two-face hypermaps.

Contribution

It introduces a novel matrix integral approach to derive generating functions for hypermaps, including recursion relations and extensions to two-face cases.

Findings

Closed-form generating functions for one-face hypermaps are obtained.

A recursion relation for these generating functions is established.

Method for extending to two-face hypermaps is outlined.

Abstract

Closed-form generating functions for counting one-face rooted hypermaps with a known number of darts by number of vertices and edges is found, using matrix integral expressions relating to the reduced density operator of a bipartite quantum system. A recursion relation for these generating functions is also found. The method for computing similar generating functions for two-face rooted hypermaps by number of vertices and edges is outlined.