The number of rooted maps on an orientable surface. (English) Zbl 0751.05052

Bender, Edward A.; Canfield, E. Rodney

J. Comb. Theory, Ser. B 53, No. 2, 293-299 (1991).

Authors’ abstract: Let \(m_ g(n)\) be the number of rooted \(n\) edged maps on an orientable surface of genus \(g>0\). The generating function of \(\rho=(1-12x)^{1/2}\) whose denominator factors completely into powers of \(\rho\), \(\rho+2\), and \(\rho+5\). We calculate \(M_ 2(x)\) and \(M_ 3(x)\). Unfortunately, we have not been able to discern a pattern in the sequence \(M_ g(x)\) from the values for \(g\leq 3\).

Reviewer: E.M.Palmer (East Lansing)

MSC:

05C30

Enumeration in graph theory

Keywords:

number of rooted maps; orientable surface; generating function

References:

[1]	Arquès, D, Relations fonctionnelles et dénombrement des Cartes pointées sur le tore, J. combin. theory ser. B, 43, 253-274, (1987) · Zbl 0628.05040
[2]	Bender, E.A; Canfield, E.R, The asymptotic number of rooted maps on a surface, J. combin. theory ser. A, 43, 244-257, (1986) · Zbl 0606.05031
[3]	Bender, E.A; Canfield, E.R; Robinson, R.W, The enumeration of maps on the torus and the projective plane, Canad. math. bull., 31, 257-271, (1988) · Zbl 0617.05036
[4]	Tutte, W.T, A census of planar maps, Canad. J. math., 15, 249-271, (1963) · Zbl 0115.17305
[5]	Walsh, T.R.S; Lehman, A.B, Counting rooted maps by genus, I, J. combin. theory ser. B, 13, 192-218, (1972) · Zbl 0228.05108
[6]	Walsh, T.R.S; Lehman, A.B; Walsh, T.R.S; Lehman, A.B, Counting rooted maps by genus, II, J. combin. theory ser. B, J. combin. theory ser. B, 14, 185-141, (1973) · Zbl 0261.05105