Polynomiality of shifted Plancherel averages and content evaluations

TL;DR

This paper proves that averages under the shifted Plancherel measure are polynomial and uses this to provide new proofs for content evaluation formulas, employing factorial Schur Q-functions as the main tool.

Contribution

It establishes polynomiality of shifted Plancherel averages and offers alternative proofs for recent content evaluation formulas using factorial Schur Q-functions.

Findings

Proved polynomiality of shifted Plancherel averages.

Provided new proofs for content evaluation formulas.

Utilized factorial Schur Q-functions as a key tool.

Abstract

The shifted Plancherel measure is a natural probability measure on strict partitions. We prove a polynomiality property for the averages of the shifted Plancherel measure. As an application, we give alternative proofs of some content evaluation formulas, obtained by Han and Xiong very recently. Our main tool is factorial Schur $Q$-functions.