### Annals of Combinatorics 1 (1997) 55-66

Enumeration of k-poles

Zhicheng Gao and Mizan Rahman

Department of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6

AMS Subject Classification: 05C10, 05C30

Abstract. A k-pole in this paper is a regular planar map with k vertices. Poles with even degrees were first enumerated by Tutte [9] in 1962 where he obtained a very simple and elegant expression. Using Brown's quadratic method, Bender and Canfield [2] derived two algebraic equations for the generating function of the poles. But the equations seem to be quite complicated for the odd degree case, and so far no progress has been seen in utilizing these equations to derive any result for the number of poles with odd degree. In this paper, we use hypergeometric functions to enumerate poles. We will show that the odd degree case is indeed very different from, and much more complicated than, the even degree case.

Keywords: enumeration, pole

References

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