A multivariate arithmetic function of combinatorial and topological significance

TL;DR

This paper studies a multivariate function called orbicyclic, which counts non-isomorphic maps on surfaces, providing a formula and linking its properties to automorphism groups of Riemann surfaces.

Contribution

It introduces and analyzes the orbicyclic function, establishing its multiplicative nature, a calculation formula, and its connection to Riemann surface automorphisms.

Findings

The orbicyclic function is multiplicative.

A simple formula for calculating the orbicyclic function is provided.

Vanishing conditions of the function align with Harvey's conditions for Riemann surface automorphisms.

Abstract

We investigate properties of a multivariate function $E(m_1,m_2,...,m_r)$, called {\it orbicyclic}, that arises in enumerative combinatorics in counting non-isomorphic maps on orientable surfaces. $E(m_1,m_2,...,m_r)$ proves to be multiplicative, and a simple formula for its calculation is provided. It is shown that the necessary and sufficient conditions for this function to vanish is equivalent to familiar Harvey's conditions that characterize possible branching data of finite cyclic automorphism groups of Riemann surfaces.