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

Abstract

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

This is the full version of a paper presented at the second Mathematics and Computer Science Conference in Versailles, France, in September 2002, in "Mathematics and Computer Science II", B. Chauvin, P. Flajolet, D. Gardy and A. Mokkadem, Editors, Trends in Mathematics, Birkhauser Verlag, Basel Switzwerland, 2002, 95--109