Cubic Pisot unit combinatorial games

Abstract

Generalized Tribonacci morphisms are defined on a three letters alphabet and generate the so-called generalized Tribonacci words. We present a family of combinatorial removal games on three piles of tokens whose set of ${\mathcal{P}}$ -positions is coded exactly by these generalized Tribonacci words. To obtain this result, we study combinatorial properties of these words like gaps between consecutive identical letters or recursive definitions of these words. β-Numeration systems are then used to show that these games are tractable, i.e., deciding whether a position is a ${\mathcal{P}}$ -position can be checked in polynomial time.