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Suppose the prime factorization of n is given by $$n=\prod_p p^v.$$

Is it possible for coefficient extraction of formal power series "on paper" where $$ F(n, k) = \left[\prod_p X_p^v\right] Z(S_k)\left(\prod_p \frac{1}{1-X_p}\right)$$ (square bracket denotes coefficient extraction of formal power series)

and $$ Z(S_n) = \frac{1}{n} \sum_{l=1}^n a_l Z(S_{n-l}) \quad\text{where}\quad Z(S_0) = 1.$$

Simankov
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    There is an example of how this can be done manually for $k$ fixed at this MSE link. You might want to add that what you appear to be doing is enumerating factorizations into multisets of factors of integers using the Lovasz recurrence for the cycle index of the symmetric group and the Polya Enumeration Theorem. – Marko Riedel May 30 '15 at 21:57
  • @MarkoRiedel thank you ! You found this question so fast) I love SE and i just can't imagine how it is possible. Every answer i asked on this site was unswered. All for free. Great! – Simankov May 30 '15 at 22:08

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