ON A CONJECTURE OF STANLEY ON JACK SYMMETRICAL FUNCTIONS

The Jack symmetric function J(lambda)(x; alpha) is a symmetric functio n with interesting properties that J(lambda)(x; 2) is a spherical func tion of the symmetric pair (GL(n, R), 0(n, R)) and that J(lambda)(x; 1 ) is the Schur function S(lambda)(x). Many interesting conjectures abo ut the combinatorial properties of J(lambda)(x; alpha) are given by St anley (1989). In this paper we give an affirmative answer to one of hi s conjectures.