scholarly journals Enumerative Formulae for Unrooted Planar Maps: a Pattern

10.37236/1841 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Valery A. Liskovets
Keyword(s):  
Rational Function ◽  
Euler Function ◽  
Planar Map ◽  
Plane Trees ◽  
Planar Maps ◽  
Map Enumeration

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.

On the Tumber of Planar Maps

1981 ◽  
Vol 33 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Nicholas C. Wormald
Keyword(s):  
Outer Region ◽  
Initial Step ◽  
Planar Map ◽  
Plane Trees ◽  
Planar Maps ◽  
Closed Plane

In a survey of methods in enumerative map theory [14], W. T. Tutte pointed out that little has been done towards enumerating unrooted maps other than plane trees. A notable exception is to be found in the work of Brown, who took the initial step in this direction by enumerating non-separable maps up to sense-preserving homeomorphisms of the plane [2]. He then took a further step, allowing sense-reversing homeomorphisms, by counting triangulations and quad-rangulations of the disc [3, 4]. In all these problems, however, there is a fixed outer region of the plane. This can be considered as a certain type of rooting of a planar map, which is normally regarded as lying on the sphere or closed plane. It is our object here to find an expression for the number of unrooted planar maps in a given set, in terms of the numbers of maps in that set which have been rooted in a special way.

scholarly journals Bijective Census and Random Generation of Eulerian Planar Maps with Prescribed Vertex Degrees

10.37236/1305 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Gilles Schaeffer
Keyword(s):  
Generating Function ◽  
Random Generation ◽  
Combinatorial Interpretation ◽  
Plane Trees ◽  
Planar Maps ◽  
Vertex Degrees

Abstract: We give a bijection between Eulerian planar maps with prescribed vertex degrees, and some plane trees that we call balanced Eulerian trees. To enumerate the latter, we introduce conjugation classes of planted plane trees. In particular, the result answers a question of Bender and Canfield and allows uniform random generation of Eulerian planar maps with restricted vertex degrees. Using a well known correspondence between 4-regular planar maps with n vertices and planar maps with n edges we obtain an algorithm to generate uniformly such maps with complexity O(n). Our bijection is also refined to give a combinatorial interpretation of a parameterization of Arquès of the generating function of planar maps with respect to vertices and faces.

scholarly journals On the Number of Planar Orientations with Prescribed Degrees

10.37236/801 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Stefan Felsner ◽  
Florian Zickfeld
Keyword(s):  
Lower Bounds ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Perfect Matchings ◽  
Asymptotic Enumeration ◽  
Combinatorial Structures ◽  
Schnyder Woods ◽  
Planar Map ◽  
Planar Maps

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with out-degrees prescribed by a function $\alpha:V\rightarrow {\Bbb N}$ unifies many different combinatorial structures, including the afore mentioned. We call these orientations $\alpha$-orientations. The main focus of this paper are bounds for the maximum number of $\alpha$-orientations that a planar map with $n$ vertices can have, for different instances of $\alpha$. We give examples of triangulations with $2.37^n$ Schnyder woods, 3-connected planar maps with $3.209^n$ Schnyder woods and inner triangulations with $2.91^n$ bipolar orientations. These lower bounds are accompanied by upper bounds of $3.56^n$, $8^n$ and $3.97^n$ respectively. We also show that for any planar map $M$ and any $\alpha$ the number of $\alpha$-orientations is bounded from above by $3.73^n$ and describe a family of maps which have at least $2.598^n$ $\alpha$-orientations.

scholarly journals A unified bijective method for maps: application to two classes with boundaries

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Olivier Bernardi ◽  
Eric Fusy
Keyword(s):  
Recursive Method ◽  
Arbitrary Size ◽  
Plane Trees ◽  
International Audience ◽  
Planar Maps ◽  
Comme Application ◽  
Natural Classes

International audience Based on a construction of the first author, we present a general bijection between certain decorated plane trees and certain orientations of planar maps with no counterclockwise circuit. Many natural classes of maps (e.g. Eulerian maps, simple triangulations,...) are in bijection with a subset of these orientations, and our construction restricts in a simple way on the subset. This gives a general bijective strategy for classes of maps. As a non-trivial application of our method we give the first bijective proofs for counting (rooted) simple triangulations and quadrangulations with a boundary of arbitrary size, recovering enumeration results found by Brown using Tutte's recursive method. En nous appuyant sur une construction du premier auteur, nous donnons une bijection générale entre certains arbres décorés et certaines orientations de cartes planaires sans cycle direct. De nombreuses classes de cartes (par exemple les eulériennes, les triangulations) sont en bijection avec un sous-ensemble de ces orientations, et notre construction se spécialise de manière simple sur le sous-ensemble. Cela donne un cadre bijectif général pour traiter les familles de cartes. Comme application non-triviale de notre méthode nous donnons les premières preuves bijectives pour l'énumération des triangulations et quadrangulations simples (enracinées) ayant un bord de taille arbitraire, et retrouvons ainsi des formules de comptage trouvées par Brown en utilisant la méthode récursive de Tutte.

Homoclinic Cycle Bifurcations in Planar Maps

2017 ◽  
Vol 27 (03) ◽  
pp. 1730012
Author(s):  
Kyohei Kamiyama ◽  
Motomasa Komuro ◽  
Kazuyuki Aihara
Keyword(s):  
Fixed Point ◽  
Lyapunov Exponent ◽  
Chaotic Attractor ◽  
Closed Curve ◽  
Bifurcation Structure ◽  
Stable And Unstable Manifolds ◽  
Planar Map ◽  
Unstable Manifolds ◽  
Planar Maps ◽  
Homoclinic Cycle

In this study, bifurcations of an invariant closed curve (ICC) generated from a homoclinic connection of a saddle fixed point are analyzed in a planar map. Such bifurcations are called homoclinic cycle (HCC) bifurcations of the saddle fixed point. We examine the HCC bifurcation structure and the properties of the generated ICC. A planar map that can accurately control the stable and unstable manifolds of the saddle fixed point is designed for this analysis and the results indicate that the HCC bifurcation depends upon a product of two eigenvalues of the saddle fixed point, and the generated ICC is a chaotic attractor with a positive Lyapunov exponent.

How to obtain a simple inverse Z- transform of a rational function of Z

1981 ◽  
Vol 128 (6) ◽  
pp. 275 ◽  
Author(s):  
Jan Staar ◽  
Joos Vandewalle
Keyword(s):  
Rational Function

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals The differential Galois group of the rational function field

2021 ◽  
Vol 381 ◽  
pp. 107605
Author(s):  
Annette Bachmayr ◽  
David Harbater ◽  
Julia Hartmann ◽  
Michael Wibmer
Keyword(s):  
Galois Group ◽  
Rational Function ◽  
Function Field ◽  
Differential Galois Group ◽  
Rational Function Field

scholarly journals The bifurcation set of a rational function via Newton polytopes

2021 ◽  
Author(s):  
Tat Thang Nguyen ◽  
Takahiro Saito ◽  
Kiyoshi Takeuchi
Keyword(s):  
Rational Function ◽  
Newton Polytopes ◽  
Bifurcation Set
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