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On the Average Number of Trees in Certain Maps

  • R. C. Mullin (a1)

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For a formal definition of “map” the reader is referred to (7, §2). The maps in this paper are rooted by specifying an orientation for one of the edges. This also specifies a root vertex, the negative end of the root, and a root face, the face on the left of the root edge. Counting is, as usual, defined on isomorphism classes.

Regular maps of even valence have been enumerated in a recent paper by Tutte. In this paper we determine the average number of trees in such maps, and include similar results for regular tri valent maps, that is, maps with three edges incident on every vertex. In the development for the latter, a formula for the number of trivalent maps with 2t vertices is produced.

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References

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1. Brown, W. G., Enumeration of triangulations of the disc, Proc. London Math. Soc, 14 (1964), 746768.
2. Etherington, M. H., Some problems in non-associative combinations, Edinburgh Math. Notes, 32 (1940), 16.
3. Harary, F., Prins, G., and Tutte, W. T., The number of plane trees, Koninkl. Nederl. Akad. Wetensch. Proc, A, 67 and Indag Math., 26 (1964), 319329.
4. Mullin, R. C., Enumeration of rooted triangular maps, Amer. Math. Monthly, 71 (1964), 10071010.
5. Mullin, R. C., On counting rooted triangular maps, Can. J. Math., 17 (1965), 373382.
6. Mullin, R. C., On the average number of Hamiltonian polygons in a rooted triangular map, Pacific J. Math, (to appear).
7. Tutte, W. T., A census of planar maps, Can. J. Math., 15 (1963), 249271.
8. Whittaker, E. T. and Watson, G. N., A Course of modern analysis (Cambridge, 1940).
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