Enumerative formulae for unrooted planar maps: a pattern
type: Item
instance of: scholarly article
title: Enumerative formulae for unrooted planar maps: a pattern
zbMATH Open document ID: 1060.05046
author: Valery A. Liskovets
published in: The Electronic Journal of Combinatorics
publication date: 2005-04-25T00:00:00Z
full work available at URL: https://eudml.org/doc/124421
full work available at URL: v11i1r88.html
review text: 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.
Mathematics Subject Classification ID: 05C30
zbMATH DE Number: 2161275
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2161275