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.