Enumeration of solid 2-trees according to edge number and edge degree distribution

Bousquet, Michel; Lamathe, Cédric

doi:10.1016/j.disc.2004.11.015

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…Formula (14 ) conforms to (1) as well. This time, however, both patterns are considered with respect to the number of polygonal blocks n, which differs (for m > 2) from the number of edges N. Formulae (14 ) and (15 ) for m = 3 also counts so-called 2-trees (resp., rooted and unrooted) with n triangles [5].…”

Section: Plane Trees Tmentioning

confidence: 99%

Enumerative Formulae for Unrooted Planar Maps: a Pattern

Liskovets¹

2004

Electron. J. Combin.

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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.

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Section: Plane Trees Tmentioning

confidence: 99%

Enumerative Formulae for Unrooted Planar Maps: a Pattern

Liskovets¹

2004

Electron. J. Combin.

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“…It appeared in [5] and [3] as an important tool in combinatorial enumeration. Suppose that a group A acts naturally (see [2, p. 393]) on a species F .…”

Section: Introduction To Species and Group Actionsmentioning

confidence: 99%

“…The quotient species (see [2, p. 159]) is defined based on group actions. It appeared in [5] and [3] as an important tool in combinatorial enumeration. Suppose that a group A acts naturally (see [2, p. 393]) on a species F .…”

Section: Introductionmentioning

confidence: 99%

Prime graphs and exponential composition of species

Li¹

2008

Journal of Combinatorial Theory, Series A

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In this paper, we enumerate prime graphs with respect to the Cartesian multiplication of graphs. We use the unique factorization of a connected graph into the product of prime graphs given by Sabidussi to find explicit formulas for labeled and unlabeled prime graphs. In the case of species, we construct the exponential composition of species based on the arithmetic product of species of Maia and Méndez, and express the species of connected graphs as the exponential composition of the species of prime graphs.

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“…Harary and Palmer [12] counted unlabeled 2-trees in 1968 (see also Harary and Palmer [13, section 3.5]) and unlabeled 2-trees were counted in a different way, using the theory of combinatorial species, by Fowler et al in [10]. Many variations of 2-trees have also been counted [4,8,12,14,15,16].…”

Section: Figure 1 a 2-treementioning

confidence: 99%