  The chromatic symmetric function of a graph is a generalization of the
chromatic polynomial. The key motivation for studying the structure of a
chromatic symmetric function is to answer positivity conjectures by Stanley in
1995 and Gasharov in 1996. As a symmetric function one can write the chromatic
symmetric function in the basis of Schur functions. In this paper we address
the positivity of the Schur coefficients when the parameter of the Schur
function is a hook shape. Furthermore, when a graph is the incomparability
graph of a poset with specific properties we construct a correspondence between
acyclic orientations of the graph and P-tableaux with a hook shape.
