On a partition identity (Q788723)

Two proofs are given for the identity \(\prod a_1!a_2!\cdots a_n!=\prod 1^{a_1}2^{a_2}\cdots n^{a_n}\), where both products range over \(\sum ia_i=n\), i.e. over all partitions of \(n\). They yield a simple proof of the known formula, \(\det^2T(n) = \left[\prod 1^{a_1}2^{a_2}\cdots n^{a_n}\right]^2\), where \(T(n)\) is the matrix formed by the character table of \(S_n\). Finally a sufficient condition is given so that the permanent of \(T(n)\) is zero.