In this paper we consider the sequence whose n th term is the number of h-vectors of length n. We show that the n th term of this sequence is bounded above by the n th Fibonacci number and bounded below by the number of integer partitions of n into distinct parts. Further we show embedded sequences that directly relate to integer partitions.

We study the picture space X d (G) of all embeddings of a finite graph G as point-andline arrangements in an arbitrary-dimensional projective space, continuing previous work on the planar case. The picture space admits a natural decomposition into smooth quasiprojective subvarieties called cellules, indexed by partitions of the vertex set of G, and the irreducible components of X d (G) correspond to cellules that are maximal with respect to a partial order on partitions that is in general weaker than refinement. We study both general properties of this partial order and its characterization for specific graphs. Our results include complete combinatorial descriptions of the irreducible components of the picture spaces of complete graphs and complete multipartite graphs, for any ambient dimension d. In addition, we give two graph-theoretic formulas for the minimum ambient dimension in which the directions of edges in an embedding of G are mutually constrained.

The slope variety of a graph is an algebraic set whose points correspond to drawings of a graph. A complement-reducible graph (or cograph) is a graph without an induced four-vertex path. We construct a bijection between the zeroes of the slope variety of the complete graph on n vertices over F 2 , and the complement-reducible graphs on n vertices.