Growth function for an $n$-valued dynamics

This article answers the question of V.M. Buchstaber about the growth function of a particular $n$-valued group. This question is closely related to discrete integrable systems. In this paper, we will find a formula for the growth function in the case when $n$ is prime. In addition, we will prove a polynomial asymptotic estimate for the growth function in the general case. Finally, we will pose new problems and hypotheses about growth functions.