Plancherel averages: Remarks on a paper by Stanley

TL;DR

This paper discusses polynomiality of averages of functions over Young diagrams under Plancherel measure, providing new proofs and extending the result to Jack deformation of the measure.

Contribution

It offers two alternative proofs of Stanley's polynomiality result and generalizes it to Jack deformation of the Plancherel measure.

Findings

Confirmed polynomial dependence of averages on n

Provided two new proofs of Stanley's result

Extended polynomiality to Jack deformation

Abstract

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.