Enumerative Formulae for Unrooted Planar Maps: a Pattern

Abstract

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 $t\mid n,\,t\! < \!n.$ 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.