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.

Topological enumeration of complex polynomial vector fields

“…This result can also be found in [Lis04]. Notice also that we obtain the same result in the case of complex polynomial vector fields where all equilibrium points are centers.…”

“…Only main arguments of this method will be given, for more details see [Lis96] and [Lis98]. The interested reader can also see [Lis96] for an application and [Lis04] for some results obtained thanks to that method. Notice that results we use here are extensions of results proved by V.A.Liskovets because we work with a generalized concept of maps, due to the presence of half-edges, but demonstrations of these results are the same.…”