Counting 5-connected planar triangulations (Q2746482)

A rooted map on the sphere is \(M\)-type if it contains neither loops nor multiple edges, and if all interior faces have degrees 3 or 4. Defining \(m_{a,b,c}\) to be the number of \(M\)-type maps with \(a\) interior triangular faces, \(b\) interior quadrangular faces, and a root face of degree \(c\), the authors determine an equation satisfied by the generating function \(M(x,y,z)=\sum m_{a,b,c}x^ay^bz^c\). The equation involves generating functions \(M_3(x,y)\) and \(M_4(x,y)\), which respectively enumerate \(M\)-type maps with a triangular and a quadrangular root face. The presence of \(M_3\) and \(M_4\) in the equation prevents solution of the equation by ``the standard quadratic method''. Instead, the authors apply a technique introduced by \textit{E. A. Bender} and \textit{E. R. Canfield} [The number of degree-restricted rooted maps on the sphere. SIAM J. Discrete Math. 7, No. 1, 9-15 (1994; Zbl 0794.05048)], based on a theorem of the reviewer [Math. Ann. 158, 82-89 (1965; Zbl 0136.02503), Math. Ann. 170, 327-333 (1967; Zbl 0144.03502)] to ultimately determine the generating function for the number \(t_n\) of rooted 5-connected planar triangulations with \(2n\) faces. It is shown in Theorem 2 that NEWLINE\[NEWLINEt_n=\frac{3c_3}{4\sqrt\pi}n^{-\frac 52}w_0^{-n}\left(1+O\left(\frac 1n\right) \right) \quad \text{as}\quad n\to\infty,NEWLINE\]NEWLINE where \(w_0\approx 0.24775354\) and \(c_3\approx 0.00067543010\). They apply a result of L. B. Richmond and the third author [cf. \textit{L. B. Richmond} and \textit{N. C. Wormald}, Almost all maps are asymmetric. J. Comb. Theory, Ser. B 63, No. 1, 1-7 (1995; Zbl 0820.05017)] to prove that the number of unrooted 5-connected planar triangulations with \(2n\) faces is asymptotic to \(\frac{c_3}{16\sqrt\pi}n^{-\frac 72}w_0^{-n}\) as \(n\to\infty\), for the same constants \(c_3\) and \(w_0\).