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Maps in locally orientable surfaces and integrals over real symmetric surfaces

Open Access article
 Printed: Oct 1997
  • I. P. Goulden
  • D. M. Jackson
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The genus series for maps is the generating series for the number of rooted maps with a given number of vertices and faces of each degree, and a given number of edges. It captures topological information about surfaces, and appears in questions arising in statistical mechanics, topology, group rings, and certain aspects of free probability theory. An expression has been given previously for the genus series for maps in locally orientable surfaces in terms of zonal polynomials. The purpose of this paper is to derive an integral representation for the genus series. We then show how this can be used in conjunction with integration techniques to determine the genus series for monopoles in locally orientable surfaces. This complements the analogous result for monopoles in orientable surfaces previously obtained by Harer and Zagier. A conjecture, subsequently proved by Okounkov, is given for the evaluation of an expectation operator acting on the Jack symmetric function. It specialises to known results for Schur functions and zonal polynomials.
MSC Classifications: 05C30, 05A15, 05E05, 15A52 show english descriptions Enumeration in graph theory
Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
Symmetric functions and generalizations
Random matrices
05C30 - Enumeration in graph theory
05A15 - Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
05E05 - Symmetric functions and generalizations
15A52 - Random matrices

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