  We prove necessary conditions for certain elementary symmetric functions,
$e_\lambda$, to appear with nonzero coefficient in Stanley's chromatic
symmetric function as well as in the generalization considered by Shareshian
and Wachs. We do this by first considering the expansion in the monomial or
Schur basis and then performing a basis change. Using the former, we make a
connection with two fundamental graph theory invariants, the independence and
clique numbers. This allows us to prove nonnegativity of three-column
coefficients for all natural unit interval graphs. The Schur basis permits us
to give a new interpretation of the coefficient of $e_n$ in terms of tableaux.
We are also able to give an explicit formula for that coefficient.
