Specifying 2-trees

Fowler, T.B.; Gessel, Ira M.; Labelle, Gilbert; Leroux, Pierre

doi:10.1007/978-3-662-04166-6_18

Cited by 4 publications

(10 citation statements)

References 5 publications

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“…We first give the following result, which is a consequence of the classical theorem of Bender (see [3]) and is inspired from the approach of Fowler et al for 2-trees (see [6,7]). Proposition 16.…”

Section: Asymptoticsmentioning

confidence: 96%

“…More recently, in 2000, Fowler, Gessel, Labelle and Leroux [7,8], have proposed some new functional equations for the class of (ordinary) 2-trees, which yield recurrences and asymptotic formulas for their unlabelled enumeration. Their approach, which is based on the theory of combinatorial species of Joyal (see [12,4]), is more structural, replacing a potential dissimilarity characteristic formula for each individual 2-tree by a Dissymmetry Theorem for the species of 2-trees.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Asymptoticsmentioning

confidence: 96%

Section: Introductionmentioning

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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“…We will use two dissymmetry formulas, analogous to the case of classical 2-trees (see Fowler and al. in [6,7]); the same proof applies in the case of plane and planar 2-trees and is omitted.…”

Section: Proposition 1 the Molecular Expansion Of The Species A = A(mentioning

confidence: 99%

“…We follow the approach of Fowler and al. in [6,7] for general 2-trees. However, we go further here, giving explicitly the molecular expansion of plane and planar 2-trees, which could not be done in the general case.…”

Section: Introductionmentioning

confidence: 99%

“…a π = a π (X) = 1 + X + E 2 (X) + X 3 + XC 3 (X) + 2E 2 (X 2 ) + X 4 + 6X 5 + · · · (5) Figure 4: First terms of the molecular expansion of the species a π of plane 2-trees a p = a p (X) = 1 + X + E 2 (X) + XE 2 (X) + XE 3 (X) + 2E 2 (X 2 ) + 2X 5 + 2XE 2 (X 2 ) + X 2 E 2 (X 2 ) + · · · + P bic 4 (X, X) + · · · + XC 3 (X 2 ) + · · · + XP bic 6 (X, X) + · · · . (6) . .…”

Section: Introductionmentioning

confidence: 99%

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A classification of plane and planar 2-trees

Labelle

Lamathe

Leroux

2003

Theoretical Computer Science

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We present new functional equations for the species of plane and of planar (in the sense of Harary and Palmer, 1973) 2-trees and some associated pointed species. We then deduce the explicit molecular expansion of these species, i.e. a classification of their structures according to their stabilizers. There result explicit formulas in terms of Catalan numbers for their associated generating series, including the asymmetry index series. This work is closely related to the enumeration of polyene hydrocarbons of molecular formula C n H n+2 .

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