Labelled and unlabelled enumeration of $k$-gonal 2-trees

Details Labelled and unlabelled enumeration of $k$-gonal 2-trees Labelled and unlabelled enumeration of $k$-gonal 2-trees

2003-12-23

arxiv.org/abs/math/0312424v1

Journal of Combinatorial Theory, Series A, 106 (2004), 193-219.

Mathematics

Combinatorics

This is the full version of a paper presented at the second Mathematics and Computer Science Conference in Versailles, France,

Scientific paper

In this paper, we generalize 2-trees by replacing triangles by quadrilaterals, pentagons or $k$-sided polygons ($k$-gons), where $k\geq 3$ is fixed. 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.

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