Enumerative Formulae for Unrooted Planar Maps: a Pattern

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 φ ( n t ) αrt (k t t ) , where φ(m) is the Euler function, k and r are integer constants and α 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.