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Zetas and moments of finite group actions. (English) Zbl 1063.11028

Summary: 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 \({\mathfrak S}_\infty\).

MSC:

11M41 Other Dirichlet series and zeta functions
20C32 Representations of infinite symmetric groups
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