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

Received November 8, 1996

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


1.  G.E. Andrews, D.M. Jackson, and T.I. Visentin, A hypergeometric analysis of the genus series for a class of 2-cell embeddings in orientable surfaces, SIAM J. Math. Anal. 25 (1994) 243–255.

2.  E.A. Bender and E.R. Canfield, The number of degree restricted rooted maps on the sphere, SIAM J. Discrete Math. 7 (1994) 9–15.

3.  A. Erdelyo et al., Eds., Higher Transcendental Functions 1, McGraw-Hill, 1953.

4.  Z.C. Gao, The number of rooted maps with a fixed number of vertices, Ars Combinatoria 35 (1993) 151–159.

5.  G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia Math. Appl. 35, Cambridge University Press, Cambridge, 1990.

6.  D.M. Jackson and T.I. Visentin, A character theoretic approach to embeddings of rooted maps in an orientable surface of given genus, Trans. Amer. Math. Soc. 322 (1990) 343–363.

7.  D.M. Jackson and T.I. Visentin, A formulation for the genus series for regular maps, J. Combin. Theory Ser. A 74 (1996) 14–32.

8.  R.P. Stanley, Enumerative Combinatorics, Vol. II, preprint, 1994.

9.  W.T. Tutte, A census of slicings, Canad. J. Math. 14 (1962) 708–722.

10.  W.T. Tutte, A census of planar maps, Canad. J. Math. 15 (1963) 249–271.

11.  T. Visentin, A character theoretic approach to the study of combinatorial problem of maps in orientable surfaces, Ph. D. Thesis, University of Waterloo, Canada, 1989.

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