On the combinatorial classification of nondegenerate configurations in the plane

Abstract

AbstractWe classify nondegenerate plane configurations by attaching, to each such configuration of n points, a periodic sequence of permutations of {1, 2, …, n} which satisfies some simple conditions; this classification turns out to be appropriate for questions involving convexity. In 1881 Perrin stated that every sequence satisfying these conditions was the image of some plane configuration. We show that this statement is incorrect by exhibiting a counterexample, for n = 5, and prove that for n ⩽ 5 every sequence essentially distinct from this one is realized geometrically by giving a complete classification of configurations in these cases; there is 1 combinatorial equivalence class for n = 3, 2 for n = 4, and 19 for n = 5. We develop some basic notions of the geometry of “allowable sequences” in the course of proving this classification theorem. Finally, we state some results and an open problem on the realizability question in the general case