  Stanley introduced expressions for the normalized characters of the symmetric
group and stated some positivity conjectures for these expressions. Here, we
give an affirmative partial answer to Stanley's positivity conjectures about
the expressions using results on Kerov polynomials. In particular, we use new
positivity results by Goulden and the present author. We shall see that the
generating series $C(t)$ introduced by them is critical to our discussion.
