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Enumerating rooted simple planar maps

Enumerating rooted simple planar maps The main purpose of this paper is to find the number of combinatorially distinct rooted simple planar maps, i.e., maps having no loops and no multi-edges, with the edge number given. We have obtained the following results. 1. The number of rooted boundary loopless planar [m, 2]-maps. i.e., maps in which there are no loops on the boundaries of the outer faces, and the edge number ism, the number of edges on the outer face boundaries is 2, is $$G_m^N \left\{ {\begin{array}{*{20}c} {1,} \\ {4 \cdot 3^{^{m - 3\frac{{(7m + 4) (2m - 3)!}}{{\left( {m - 2} \right)! \left( {m + 2} \right)!}},} } } \\\end{array}} \right.\begin{array}{*{20}c} {if m = 1,2;} \\ {m \ge 3,} \\\end{array}$$ form≥1.G 0 N =0. 2. The number of rooted loopless planar [m, 2]-maps is $$G_m^{NL} \left\{ {\begin{array}{*{20}c} {0,} \\ {\frac{{6 \cdot (4m - 3)}}{{\left( {m - 1} \right)!\left( {3m} \right)!}},} \\\end{array}} \right.\begin{array}{*{20}c} {if m = 0;} \\ {if m \ge 1.} \\\end{array}$$ 3. The number of rooted simple planar maps withm edgesH m s satisfies the following recursive formula: $$G_m^{NL} \left\{ {\begin{array}{*{20}c} {H_m^3 = H_m^{NL} - \sum\limits_{i = 1}^{m - 1} {\gamma (i, m)H_i^3 , m \ge 2;} } \\ {H_0^3 = H_1^3 = 1,} \\\end{array}} \right.$$ whereH m NL is the number of rooted loopless planar maps withm edges given in [2]. 4. In addition, γ(i, m),i≥1, are determined by $$G_m^{NL} \left\{ {\begin{array}{*{20}c} {\gamma (i,m) = \sum\limits_{i = 1}^{m - 1} {\frac{{(4j)!}}{{\left( {3j + 1} \right)!j!}}\frac{{m - j}}{{m - i}}\gamma (i,m - j), m \ge i + 1;} } \\ {\gamma (i, i) = 1} \\\end{array}} \right.$$ form≥i. γ(i, j)=0, wheni>j. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Enumerating rooted simple planar maps

Acta Mathematicae Applicatae Sinica , Volume 2 (2) – Apr 6, 2005

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Publisher
Springer Journals
Copyright
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/BF01539481
Publisher site
See Article on Publisher Site

Abstract

The main purpose of this paper is to find the number of combinatorially distinct rooted simple planar maps, i.e., maps having no loops and no multi-edges, with the edge number given. We have obtained the following results. 1. The number of rooted boundary loopless planar [m, 2]-maps. i.e., maps in which there are no loops on the boundaries of the outer faces, and the edge number ism, the number of edges on the outer face boundaries is 2, is $$G_m^N \left\{ {\begin{array}{*{20}c} {1,} \\ {4 \cdot 3^{^{m - 3\frac{{(7m + 4) (2m - 3)!}}{{\left( {m - 2} \right)! \left( {m + 2} \right)!}},} } } \\\end{array}} \right.\begin{array}{*{20}c} {if m = 1,2;} \\ {m \ge 3,} \\\end{array}$$ form≥1.G 0 N =0. 2. The number of rooted loopless planar [m, 2]-maps is $$G_m^{NL} \left\{ {\begin{array}{*{20}c} {0,} \\ {\frac{{6 \cdot (4m - 3)}}{{\left( {m - 1} \right)!\left( {3m} \right)!}},} \\\end{array}} \right.\begin{array}{*{20}c} {if m = 0;} \\ {if m \ge 1.} \\\end{array}$$ 3. The number of rooted simple planar maps withm edgesH m s satisfies the following recursive formula: $$G_m^{NL} \left\{ {\begin{array}{*{20}c} {H_m^3 = H_m^{NL} - \sum\limits_{i = 1}^{m - 1} {\gamma (i, m)H_i^3 , m \ge 2;} } \\ {H_0^3 = H_1^3 = 1,} \\\end{array}} \right.$$ whereH m NL is the number of rooted loopless planar maps withm edges given in [2]. 4. In addition, γ(i, m),i≥1, are determined by $$G_m^{NL} \left\{ {\begin{array}{*{20}c} {\gamma (i,m) = \sum\limits_{i = 1}^{m - 1} {\frac{{(4j)!}}{{\left( {3j + 1} \right)!j!}}\frac{{m - j}}{{m - i}}\gamma (i,m - j), m \ge i + 1;} } \\ {\gamma (i, i) = 1} \\\end{array}} \right.$$ form≥i. γ(i, j)=0, wheni>j.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Apr 6, 2005

References