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Maps in Locally Orientable Surfaces, the Double Coset Algebra, and Zonal Polynomials

Published online by Cambridge University Press:  20 November 2018

I. P. Goulden  and
D. M. Jackson
I. P. Goulden
Affiliation:
Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario N2L 3G1
D. M. Jackson
Affiliation:
Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario N2L 3G1
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Abstract

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The genus series is the generating series for the number of maps (inequivalent two-cell embeddings of graphs), in locally orientable surfaces, closed and without boundary, with respect to vertex- and face-degrees, number of edges and genus. A hypermap is a face two-colourable map. An expression for the genus series for (rooted) hypermaps is derived in terms of zonal polynomials by using a double coset algebra in conjunction with an encoding of a map as a triple of matchings. The expression is analogous to the one obtained for orientable surfaces in terms of Schur functions.

Keywords

05E0505A1557M15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 3 , 01 June 1996 , pp. 569 - 584
Copyright
Copyright © Canadian Mathematical Society 1996

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