Enumerative formulae for unrooted planar maps: a pattern (Q1773160)

Summary: We present uniformly available simple enumerative formulae for unrooted planar \(n\)-edge maps (counted up to orientation-preserving isomorphism) of numerous classes including arbitrary, loopless, non-separable, Eulerian maps and plane trees. All the formulae conform to a certain pattern with respect to the terms of the sum over \(t| n\), \(t<n\). Namely, these terms, which correspond to non-trivial automorphisms of the maps, prove to be of the form \(\phi\left({n\over t}\right)\alpha\,r^t {k\,t\choose t}\), where \(\phi(m)\) is the Euler function, \(k\) and \(r\) are integer constants and \(\alpha\) is a constant or takes only two rational values. On the contrary, the main, ``rooted'' summand corresponding to \(t=n\) contains an additional factor which is a rational function of \(n\). Two simple new enumerative results are deduced for bicolored Eulerian maps. A collateral aim is to briefly survey recent and old results of unrooted planar map enumeration.