Enumération des 2-arbres k-gonaux

Labelle, Gilbert; Lamathe, Cédric; Leroux, Pierre

doi:10.1007/978-3-0348-8211-8_6

Cited by 5 publications

(5 citation statements)

References 7 publications

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“…See for example [9]. Inspired by formulas of Norman ([6], (18)) and Robinson ( [27], Theorem 7), we have given over the years a more structural formula which we call the Dissymmetry theorem for graphs, whose proof is remarkably simple and which can esily be adapted to various classes of tree-like structures; see [2], [3], [7], [14], [15], [16], [18], [19]. Second, as for trees, it should not be expected to obtain simple closed expressions but rather recurrence formulas for the number of unlabelled C B -structures.…”

Section: Outlinementioning

confidence: 99%

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Enumerative Problems Inspired by Mayer's Theory of Cluster Integrals

Leroux¹

2004

Electron. J. Combin.

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The basic functional equations for connected and 2-connnected graphs can be traced back to the statistical physicists Mayer and Husimi. They play an essential role in establishing rigorously the virial expansion for imperfect gases. We survey this approach and inspired by these equations, we investigate the problem of enumerating some classes of connected graphs all of whose blocks are contained in a given class B. Included are the species of Husimi graphs (B = "complete graphs"), cacti (B = "unoriented cycles"), and oriented cacti (B = "oriented cycles"). For each of these, we consider the question of their labelled or unlabelled enumeration and of their molecular expansion, according (or not) to their block-size distributions. Résumé Leséquations fonctionnelles de base pour les graphes connexes et 2-connexes remontent aux physiciens Mayer et Husimi. Ces relations sont importantes pouŕ etablir rigoureusement le développement du viriel pour les gaz imparfaits. Nous présentons cette démarche et inspirés par ces relations, nous examinons le problème du dénombrement de quelques classes de graphes connexes dont les blocs sont pris dans une famille B donnée. Cela inclut les graphes de Husimi (B = "graphes complets"), les cactus (B = "polygones"), et les cactus orientés (B = "cycles orientés"). Pour chacune de ces espèces, on s'intéresse au dénombrementétiqueté, ordinaire ou selon la distribution des tailles des blocs, au dénombrement non-étiqueté, et au développement moléculaire.

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Section: Outlinementioning

confidence: 99%

“…In particular, f (r) vanishes when r is greater than the range r 1 of the interaction potential. By substituting in the canonical partition function (15), one obtains…”

Section: The Virial Expansionmentioning

confidence: 99%

Enumerative Problems Inspired by Mayer's Theory of Cluster Integrals

Leroux¹

2004

Electron. J. Combin.

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“…Here the norm function satisfies N(ϕαβ + γ) = 1 + Nα + Nβ + Nγ where ϕαβ + γ is in Cantor normal form and ϕ denotes the binary fixed point free Veblen function. The resulting generating function for counting these trees has radius of convergence smaller than one by [10,4].…”

Section: Limit Laws For Ordinals At Least a Big As εmentioning

confidence: 99%

Monadic second order limit laws for natural well orderings

Weiermann

2020

Preprint

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By combining classical results of Büchi, some elementary Tauberian theorems and some basic tools from logic and combinatorics we show that every ordinal α with ε 0 ≥ α ≥ ω ω satisfies a natural monadic second order limit law and that every ordinal α with ω ω > α ≥ ω satisfies a natural monadic second order Cesaro limit law. In both cases we identify as usual α with the class of substructures {β : β < α}.We work in an additive setting where the norm function N assigns to every ordinal α the number of occurrrences of the symbol ω in its Cantor normal form. This number is the same as the number of edges in the tree which is canonically associated with α.For a given α with ω ≤ α ≤ ε 0 the asymptotic probability of a monadic second order formula ϕ from the language of linear orders is lim n→∞ #{β<α:N β=n∧β|=Φ} #{β<α:N β=n} if this limit exists. If this limit exists only in the Cesaro sense we speak of the Cesaro asympotic probability of ϕ.Moreover we prove monadic second order limit laws for the ordinal segments below below Γ 0 (where the norm function is extended appropriately) and we indicate how this paper's results can be extended to larger ordinal segments and even to certain impredicative ordinal notation systems having notations for uncountable ordinals. We also briefly indicate how to prove the corresponding multiplicative results for which the setting is defined relative to the Matula coding.The results of this paper concerning ordinals not exceeding ε 0 have been obtained partly in joint work with Alan R. Woods.

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“…The sequences {b n } n∈N , for k = 2, 3, 4, 5, 6, are listed in the Encyclopedia of Integer Sequences [25,26]. Respectively: A000081, for the number of rooted trees with n nodes, A005750, in relation with planted matched trees with n nodes and 2-trees, A052751, A052773, A052781, in relation with equation (19). Also, equation (9), is referenced in the Encyclopedia of Combinatorial Structures [11].…”

Section: B) A)mentioning

confidence: 99%

“…This is the full version of a paper presented at the International Colloquium on "Mathematics and Computer Science" held in Versailles, France, in September 2002 (see[19]). …”

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confidence: 99%

Labelled and unlabelled enumeration of k-gonal 2-trees

Labelle

Lamathe

Leroux

2004

Journal of Combinatorial Theory, Series A

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In this paper, we generalize 2-trees by replacing triangles by quadrilaterals, pentagons or k-sided polygons (k-gons), where kX3 is given. This generalization, to k-gonal 2-trees, is natural and is closely related, in the planar case, to some specializations of the cell-growth problem. Our goal is the labelled and unlabelled enumeration of k-gonal 2-trees according to the number n of k-gons. We give explicit formulas in the labelled case, and, in the unlabelled case, recursive and asymptotic formulas. r 2004 Elsevier Inc. All rights reserved.

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Enumération des 2-arbres k-gonaux

Cited by 5 publications

References 7 publications

Enumerative Problems Inspired by Mayer's Theory of Cluster Integrals

Enumerative Problems Inspired by Mayer's Theory of Cluster Integrals

Monadic second order limit laws for natural well orderings

Labelled and unlabelled enumeration of k-gonal 2-trees

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