Enumerating rooted loopless planar maps
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Abstract
- 1.
The equivalence between the method described by W. T. Tutte for determining parametric expressions of certain enumerating functions[6] and the one which the author used in [2] for finding the parametric expression of the generating function of rooted general planar maps dependent on the edge number, is shown.
- 2.
The number of rooted boundary loop maps, i.e., maps for each of which all the edges on the boundary of the outer face are loops, with the edge number given is found.
- 3.
The number of rooted nearly loopless planar maps, i.e., loopless maps and maps having exactly one loop which is just the rooted edge and does not form the boundary of the outer face, with given edge number is also found.
- 4.
The recursive formula satisfied by the number of rooted loopless planar maps dependent on the edge number is derived.
- 5.
In addition, the number of loop rooted maps, i.e., maps in each of which there is only one loop which is just the rooted edge, dependent on the edge number is obtained at the same time.
Keywords
Generate Function Math Application Recursive Formula Outer Face Boundary LoopPreview
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References
- [1]Brown, W. G., Enumeration of Triangulations of the Disk,Proc. London Math. Soc.,14: 3 (1964), 746–768.Google Scholar
- [2]Liu Yanpei, Enumeration of Rooted Separable Planar Maps, Research Report CORR 82-47, University of Waterloo, 1982. Also inUtilitas Math.,25 (1984), 77–94.Google Scholar
- [3]Mullin, R. C., On Counting Rooted Triangular Maps,Canad. J. Math.,17 (1965), 373–382.Google Scholar
- [4]Tutte, W.T., A Census of Planar Triangulations.Canad. J. Math.,14, (1962), 21–38.Google Scholar
- [5]Tutte, W. T., A Census of Planar Maps,Canad. J. Math.,15 (1963), 249–271.Google Scholar
- [6]Tutte, W. T., Counting Rooted Triangulations,Annals of Discrete Mathematics,12 (1982), 243–253.Google Scholar