Cesàro summation and multiplicative functions on a symmetric group

σ = κ1κ2...κω this decomposition is unique up to the order of the multiplicands. We will call a function f : Sn → C multiplicative if f(σ) = f(κ1)f(κ2)...f(κn). In what follows we will assume that the value of f on cycles depends only on the length of cycle, that is f(κ) = f̂(|κ|), where |κ| the order of cycle κ. Let mk(σ) be equal to the number of cycles in the decomposition σ whose order is equal to k. Then obviously m1(σ) + 2m2(σ) + ... + nmn(σ) = n. Thus n complex number f̂(1), f̂(2), ..., f̂(n) completely determine the value of function f on any permutation σ ∈ Sn