Difference operators for partitions under the Littlewood decomposition

Abstract

The concept of $\mathit{t}$-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 $\mathit{t}$-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 $\mathit{t}$.