Abstract

An M × N matrix is associated with each ordered k-arc in a finite projective space of order N (k = M + N + 2.) The matrix is a projective invariant for ordered k-arcs in the space. The set of these matrices is denoted by Ω. Elements of the symmetric group Sk act on ordered arcs by permuting points. This induces a definition of Sk as a group of operators on Ω, whose orbits correspond to projectively distinct unordered k-arcs. Application of theorems of Burnside and Cauchy leads to results concerning the number of orbits of k-arcs in PG(N, q) under projectivity and under collineation. A subset of Ω is defined which contains representatives of each orbit under Sk. The reduced set of “normal” matrices is used by a counting algorithm. The results of this paper are applied to counting the projectively distinct unordered k-arcs (all k) in PG(2, 11) and PG(2, 13).

Introduction

This paper is concerned with the problem of counting equivalence classes of ordered k-arcs, unordered k-arcs, and k-gons with respect to either projectivity or collineation. Taking into account the three types of arc and two equivalence relations, there are six distinct but related problems to be solved. The six problems can each be stated in terms of the orbits of ordered arcs under some group of operators. The solutions, themselves, fall into two categories: formulas for the number of orbits and algorithms for counting the orbits.