Plancherel averages: Remarks on a paper by Stanley

Let M_n stand for the Plancherel measure on Y_n, the set of Young diagrams with n boxes. A recent result of Stanley (arXiv:0807.0383) says that for certain functions G defined on the set Y of all Young diagrams, the average of G with respect to M_n depends on n polynomially. We propose two other proofs of this result together with a generalization to the Jack deformation of the Plancherel measure.