Positivity results for Stanleyʹs character polynomials

In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin. 50 (B50d) (2003) 11 pp.] the author introduces expressions for the normalized characters of the symmetric group and states some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanleyʹs positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results in Goulden and Rattan [I.P. Goulden, A. Rattan, An explicit form for Kerovʹs character polynomials, Trans. Amer. Math. Soc., in press, math.CO/0505317, November 2005]. We shall see that the generating series C(t) introduced in [I.P. Goulden, A. Rattan, An explicit form for Kerovʹs character polynomials, Trans. Amer. Math. Soc., in press, math.CO/0505317, November 2005] is critical to our discussion.