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Counting unrooted loopless planar maps. (English) Zbl 1070.05050
This paper is devoted to proving the following formula for \(L^+(n)\), the number of loopless planar maps with \(n\) edges up to an orientation-preserving isomorphism:
Theorem 1. For \(n\geq1\), \[ L^+(n)=\frac1{2n}\Biggl[ \frac{2(4n+1)}{(n+1)(3n+1)(3n+2)} \binom{4n} {n}+ \sum_{t<n,t|n}\phi \biggl(\frac nt \biggr) \binom{4t}{t}+ \begin{cases} \frac{2n}{n+1}{2n\choose \frac{n-1}2} \!&\! \text{if \(n\) is odd}\\ \binom{2n} {\frac{n-2}2} \!&\! \text{if \(n\) is even} \end{cases} \Biggr]\!, \] where \(\phi(n)\) is the Euler totient function.

MSC:
05C30 Enumeration in graph theory
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