Non-Schur-positivity of chromatic symmetric functions

David G. L. Wang, Monica M. Y. Wang

Published 2020-01-01Version 1

We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. As applications, we establish the non-Schur-positivity of some graph families. These graph families include the windmill graphs, non-balanced bipartite graphs, complete bipartite graphs, complete tripartite graphs, and the wheel graphs, when the number of vertices are not too small. We also show that the Dynkin graphs of type $D_n$ and type $E_n$ are not $e$-positive for $n>10$.