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Transactions of the American Mathematical Society

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On an integral representation for the genus series for $ 2$-cell embeddings

Author: D. M. Jackson
Journal: Trans. Amer. Math. Soc. 344 (1994), 755-772
MSC: Primary 05C10; Secondary 05C30, 20C15, 58C35
MathSciNet review: 1236224
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Abstract: An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2-cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertex-regular maps in general and, in this way, may shed light on the genus series for quadrangulations. The integral representation is used in conjunction with an approach through the group algebra, $ \mathbb{C}{\mathfrak{G}_n}$, of the symmetric group [11] to obtain a factorisation of certain Gaussian integrals.

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Article copyright: © Copyright 1994 American Mathematical Society

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