Andrew M. Baxter

Periodic Orbits for Billiards on an Equilateral Triangle

1. INTRODUCTION. The trajectory of a billiard ball in motion on a frictionless billiards table is completely determined by its initial position, direction, and speed. When the ball strikes a bumper, we assume that the angle of incidence equals the angle of reflection. Once released, the ball continues indefinitely along its trajectory with constant speed unless it strikes a vertex, at which point it stops. If the ball returns to its initial position with its initial velocity direction, it retraces its trajectory and continues to do so repeatedly; we call such trajectories periodic. Nonperiodic trajectories are either infinite or singular; in the later case the trajectory terminates at a vertex.

Automatic Generation of Theorems and Proofs on Enumerating Consecutive-Wilf Classes

This article, describes two complementary approaches to enumeration, the positive and the negative, each with its advantages and disadvantages. Both approaches are amenable to automation, and we apply it to the currently active subarea, initiated in 2003 by Sergi Elizalde and Marc Noy, of enumerating consecutive-Wilf classes (i.e. consecutive pattern-avoidance) in permutations.

Applying the cluster method to count occurrences of generalized permutation patterns

We apply ideas from the cluster method to q-count the permutations of a multiset according to the number of occurrences of certain generalized patterns, as defined by Babson and Steingrímsson. In particular, we consider those patterns with three letters and one internal dash, as well as permutation statistics composed of counting the number of occurrences of multisets of such patterns. Counting is done via recurrences which simplify in the case of permutations. A collection of Maple procedures implementing these recurrences accompanies the article.