Zetas and moments of finite group actions

Abstract

We introduce and study two kinds of zeta functions $\zeta(u; G,X)$ and $Z(u; G,X)$ as well as moments $m(k; G,X)$ attached to a given finite group action $G \curvearrowright X$. We show that zeta functions determine the moments, and moments determine the multiple transitivity of the action. In the symmetric group case we give an explicit formula of moments and calculate zeta functions of the infinite symmetric group $\mathfrak{S}_{\infty}$.