The Penney's Game with Group Action

Abstract

We generalize word avoidance theory by equipping the alphabet A with a group action. We call equivalence classes of words patterns. We extend the notion of word correlation to patterns using group stabilizers. We extend known word avoidance results to patterns. We use these results to answer standard questions for the Penney's game on patterns and show non-transitivity for the game on patterns as the length of the pattern tends to infinity. We also analyze bounds on the pattern-based Conway leading number and expected wait time, and further explore the game under the cyclic and symmetric group actions.