Counting unrooted maps on the plane
type: Item
instance of: scholarly article
title: Counting unrooted maps on the plane
zbMATH Open document ID: 1112.05054
author: Valery A. Liskovets
author: Timothy R. S. Walsh
published in: Advances in Applied Mathematics
publication date: 2007-01-16T00:00:00Z
review text: A plane map is a 2-cell imbedding of a connected multigraph (loops and multiple edges allowed) on the sphere, with a distinguished face. The map is rooted by also distinguishing an oriented edge; an unrooted plane map is an equivalence class of plane maps under orientation-preserving homeomorphism. In a series of three papers [Can. J. Math. 35, 417--435 (1983; Zbl 0519.05041); Discrete Math. 282, No. 1--3, 209--221 (2004; Zbl 1051.05049) and Eur. J. Comb. 26, No. 5, 651--663 (2005; Zbl 1070.05050)], the authors enumerated unrooted planar (no distinguished face) \(n\)-edge maps of various classes, including all maps, non-separable maps, Eulerian maps, and loopless maps. In the present paper, they employ the same technique to get closed formulae for counting unrooted plane maps of all these classes and their duals. In contrast to the rooted case, where the corresponding formulae are all sum-free, the formulae obtained for unrooted maps contain a sum over the divisors of \(n\). Also counted are unrooted two-vertex plane maps.
Mathematics Subject Classification ID: 05C30
zbMATH DE Number: 5116340
zbMATH Keywords: rooted planar map
zbMATH Keywords: unrooted plane map
zbMATH Keywords: quotient map
zbMATH Keywords: sum-free formula
zbMATH Keywords: outside face
reviewed by: Arthur T. White
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: j.aam.2005.03.006
OpenAlex ID: W2043514463
Wikidata QID: Q60692082
cites work: Enumeration of planar two-face maps
cites work: Enumeration of Non-Separable Planar Maps
cites work: Q5333061
cites work: Q4552156
cites work: Une nouvelle demonstration combinatoire des formules d'inversion de Lagrange
cites work: Q3669450
cites work: Q3699715
cites work: Enumerative formulae for unrooted planar maps: a pattern
cites work: The Enumeration of Non-Isomorphic 2-Connected Planar Maps
cites work: Enumeration of Eulerian and unicursal planar maps
cites work: Counting unrooted loopless planar maps
cites work: Q4875364
cites work: Q4239067
cites work: The On-Line Encyclopedia of Integer Sequences
cites work: A Census of Planar Maps
cites work: The number of plane trees
cites work: Hypermaps versus bipartite maps
cites work: Efficient enumeration of sensed planar maps
cites work: Counting rooted maps by genus. III: Nonseparable maps
DOI: 10.1016/J.AAM.2005.03.006
description: scientific article; zbMATH DE number 5116340