A combinatorial formula for the Schur coefficients of chromatic symmetric functions

We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a graph, especially when Stanley's stable partition method does not work. As applications, we completely characterize Schur positive complete tripartite graphs. We show that any squid graph obtained by attaching n pendent edges to a common vertex on the cycle C_m is not Schur positive if m≠2n−1, and that any pineapple graph obtained by attaching m pendent edges to a common vertex on the complete graph K_n is not Schur positive if n≤2^m−2.