Difference operators for partitions under the Littlewood decomposition

TL;DR

This paper introduces $t$-difference operators for partitions to generalize Stanley's theorem, utilizing a Littlewood decomposition-based measure to analyze contents and hook lengths in Young diagrams.

Contribution

It extends the concept of difference operators and measures on partitions, providing new insights into contents and hook lengths via Littlewood decomposition.

Findings

Generalization of Stanley's theorem on polynomiality

Introduction of $t$-difference operators for partitions

Analysis of contents and hook lengths in congruence classes

Abstract

The concept of $t$-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of $t$-hooks. It is well-known that the hook lengths of multiples of $t$ can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes modulo $t$.