A multivariate arithmetic function of combinatorial and topological significance

Details A multivariate arithmetic function of combinatorial and topological significance A multivariate arithmetic function of combinatorial and topological significance

2009-03-30

arxiv.org/abs/0903.4965v2

Integers, 10 (2010), A12, 155-177 (the last remark has been strengthened)

Mathematics

Number Theory

17 pages; new Remark 3.10 and Concluding remarks; additional references; minor improvements. Final version to appear in INTEGE

Scientific paper

10.1515/INTEG.2010.012

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.

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