Enumeration of 2-trees with directed cells

<jats:p>We consider the problem of enumerating 2-trees with oriented cells up to isomorphism. A 2-tree is a simple graph obtained from K3 by the iterative addition of vertices connected to the ends of some edge. The cells of a 2-tree are oriented if in each cell (triangle) the vertices are independently labeled 1, 2, 3. The cells are partially oriented if one vertex is independently labeled in each cell. Using the Redfield - Polya theorem and dissymmetry lemma, we compute generating functions for the number of 2-trees with oriented and partially oriented cells.</jats:p>