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We develop a composition method to unearth positive $e_I$-expansions ofchromatic symmetric functions $X_G$, where the subscript $I$ stands forcompositions rather than integer partitions. Using this method, we derivepositive and neat $e_I$-expansions for the chromatic symmetric functions oftadpoles, barbells and generalized bulls, and establish the $e$-positivity ofhats. We also obtain a compact ribbon Schur analog for the chromatic symmetricfunction of cycles.

A composition method for neat formulas of chromatic symmetric functions