Point configurations on the projective line over a finite field

Abstract

We study n-point configurations in $${\mathbb{P}^1(\mathbb{F}_q)}$$ modulo projective equivalence. For n = 4 and 5, a complete classification is given, along with the numbers of such configurations with a given symmetry group. Using Polya’s coloring theorem, we investigate the behavior of the numbers C(n, q) of classes of n-configurations resp. C spec(n, q) of classes with nontrivial symmetry group. Both are described by rational polynomials in q which depend on q modulo $${\lambda(n) = {\rm lcm} \{m \in \mathbb{N} | m \leq n\}}$$ .