  For a $(3+1)$-free poset $P$, we define a hybrid of $P$-tableaux and
cylindric tableaux called cylindric $P$-tableaux. We introduce $P$-analogs of
cylindric Schur functions, defined by a determinantal formula, and prove that
they are the weight generating functions of cylindric $P$-tableaux. We deduce
that certain sums of the $e$-expansion coefficients of the chromatic symmetric
function $X_{inc(P)}$ are positive. This improves on Gasharov's theorem on the
Schur positivity of $X_{inc(P)}$ and gives further evidence for the
Stanley-Stembridge conjecture.
