  A well-known result of Stanley's shows that given a graph $G$ with chromatic
symmetric function expanded into the basis of elementary symmetric functions as
$X_G = \sum c_{\lambda}e_{\lambda}$, the sum of the coefficients $c_{\lambda}$
for $\lambda$ with exactly $k$ parts is equal to the number of acyclic
orientations of $G$ with exactly $k$ sinks.
  However, more is known. The $sink$ $sequence$ of an acyclic orientation of
$G$ is a tuple $(s_1,\dots,s_k)$ such that $s_1$ is the number of sinks of the
orientation, and recursively each $s_i$ with $i > 1$ is the number of sinks
remaining after deleting the sinks contributing to $s_1,\dots,s_{i-1}$.
Equivalently, the sink sequence gives the number of vertices at each level of
the poset induced by the acyclic orientation.
  A lesser-known follow-up result of Stanley's determines certain cases in
which we can find a sum of $e$-basis coefficients that gives the number of
acyclic orientations of $G$ with a given partial sink sequence. Of interest in
its own right, this result also admits as a corollary a simple proof of the
$e$-positivity of $X_G$ when the stability number of $G$ is $2$.
  In this paper, we prove a vertex-weighted generalization of this follow-up
result, and conjecture a stronger version that admits a similar combinatorial
interpretation for a much larger set of $e$-coefficient sums of chromatic
symmetric functions. In particular, the conjectured formula would give a
combinatorial interpretation for the sum of the coefficients $c_{\lambda}$ with
prescribed values of $\lambda_1'$ and $\lambda_2'$ for any unweighted claw-free
graph.
