On a conjecture of Stanley on Jack symmetric functions

Abstract

The Jack symmetric function J_λ(x;α) is a symmetric function with interesting properties that J_λ(x;2) is a spherical function of the symmetric pair (GL(n,$\text{R}$), O(n,$\text{R}$) and that J_λ(x;1) is the Schur function S_λ(x). Many interesting conjectures about the combinatorial properties of J_λ(x;α) are given by Stanley (1989). In this paper we give an affirmative answer to one of his conjectures.