# Enumerating rooted loopless planar maps

Enumerating rooted loopless planar maps This paper provides the following results. 1. The equivalence between the method described by W. T. Tutte for determining parametric expressions of certain enumerating functions and the one which the author used in  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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Enumerating rooted loopless planar maps

, Volume 2 (1) – Apr 26, 2005
13 pages      /lp/springer-journals/enumerating-rooted-loopless-planar-maps-T0qGsUQ9kB
Publisher
Springer Journals
Copyright © 1985 by Science Press and D. Reidel Publishing Company
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF01666515
Publisher site
See Article on Publisher Site

### Abstract

This paper provides the following results. 1. The equivalence between the method described by W. T. Tutte for determining parametric expressions of certain enumerating functions and the one which the author used in  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.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Apr 26, 2005

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