On a conjecture of Stanley on Jack symmetric functions

Koike, K., On a conjecture of Stanley on Jack symmetric functions, Discrete Mathematics 115 (1993) 211-216. The Jack symmetric function J,(x; G() is a symmetric function with interesting properties that J,(x; 2) is a spherical function of the symmetric pair (GL(n, FQ O(n, [w)) and that J,(x; 1) is the Schur function S,(x). Many interesting conjectures about the combinatorial properties of J,(x;cc) are given by Stanley (1989). In this paper we give an affirmative answer to one of his conjectures. 0. Introduction In [6] Stanley gave a series of interesting conjectures concerning Jack symmetric functions. In this paper we give an affirmative answer to one of his conjectures, which is as follows. Conjecture 0.1. Here J,(x;a) is the Jack symmetric function, with parameter a corresponding to a partition 1” (see Section 1 and also [6] for the precise definition) and SA(x) is the Schur function corresponding to a partition I and (a),:= a(a 1) (a 2) ... (a r + 1) denotes the lower factorial. Correspondence to: Kazuhiko Koike, Department of Mathematics, Aoyama Gakuin University, Setagaya-ku, Tokyo 157, Japan. * Partially supported by Grant Aid for Scientific Research. 0012-365X/93/$06.00