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How to prove $\prod_{\lambda\vdash n}\prod_im_i(\lambda)!=\prod_{\lambda\vdash n}1^{m_1(\lam...
=\prod\limits_{\lambda\vdash n}1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots$$ It is clear that the right side of the identity is the product of all parts of all partitions of $n$. …