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A Multivariate Arithmetic Function of Combinatorial and Topological Significance

Published/Copyright: March 19, 2010
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Integers
From the journal Volume 10 Issue 1

Abstract

We investigate properties of a multivariate function E (m1, m2, …, mr), called orbicyclic, that arises in enumerative combinatorics in counting non-isomorphic maps on orientable surfaces. E(m1, m2, …, mr) 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 are equivalent to familiar Harvey's conditions that characterize possible branching data of finite cyclic automorphism groups of Riemann surfaces.

Keywords.: Ramanujan trigonometric sum; von Sterneck function; Jordan function; multiplicative function; map enumeration; unrooted map on orientable surface; cyclic group action; automorphism of Riemann surface; Harvey's conditions; orbifold; order-preserving epimorphism
Received: 2009-05-06
Accepted: 2009-12-19
Published Online: 2010-03-19
Published in Print: 2010-March

© de Gruyter 2010

Articles in the same Issue

  1. Congruences for Hyper m-ary Overpartition Functions
  2. A Note on Fibonacci-Type Polynomials
  3. Non-Regularity of ⌊α + logk n⌋
  4. Lerch's Theorems over Function Fields
  5. On Normal Numbers and Powers of Algebraic Numbers
  6. Volume as a Measure of Approximation for the Jacobi–Perron Algorithm
  7. Combinatorics of Integer Partitions in Arithmetic Progression
  8. Analogues of Jacobi's Two-Square Theorem: An Informal Account
  9. Convolution Identities for Stirling Numbers of the First Kind
  10. Pattern Occurrence in the Dyadic Expansion of Square Root of Two and an Analysis of Pseudorandom Number Generators
  11. Place-Difference-Value Patterns: A Generalization of Generalized Permutation and Word Patterns
  12. A Multivariate Arithmetic Function of Combinatorial and Topological Significance
https://doi.org/10.1515/integ.2010.012
Keywords for this article
Ramanujan trigonometric sum; von Sterneck function; Jordan function; multiplicative function; map enumeration; unrooted map on orientable surface; cyclic group action; automorphism of Riemann surface; Harvey's conditions; orbifold; order-preserving epimorphism

Articles in the same Issue

  1. Congruences for Hyper m-ary Overpartition Functions
  2. A Note on Fibonacci-Type Polynomials
  3. Non-Regularity of ⌊α + logk n⌋
  4. Lerch's Theorems over Function Fields
  5. On Normal Numbers and Powers of Algebraic Numbers
  6. Volume as a Measure of Approximation for the Jacobi–Perron Algorithm
  7. Combinatorics of Integer Partitions in Arithmetic Progression
  8. Analogues of Jacobi's Two-Square Theorem: An Informal Account
  9. Convolution Identities for Stirling Numbers of the First Kind
  10. Pattern Occurrence in the Dyadic Expansion of Square Root of Two and an Analysis of Pseudorandom Number Generators
  11. Place-Difference-Value Patterns: A Generalization of Generalized Permutation and Word Patterns
  12. A Multivariate Arithmetic Function of Combinatorial and Topological Significance
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