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We consider a linear relation which expresses Stanley's chromatic symmetricfunction for a poset in terms of the chromatic symmetric functions of someclosely related posets, which we call the modular law. By applying this in thecontext of (3+1)-free posets, we are able to reduce Stanley and Stembridge'sconjecture that the chromatic symmetric functions of all (3+1)-free posets aree-positive to the case of (3+1)-and-(2+2)-free posets, also known as unitinterval orders. In fact, our reduction can be pushed further to a much smallerclass of posets, for which we have no satisfying characterization. We alsoobtain a new proof of the fact that all 3-free posets have e-positive chromaticsymmetric functions.

A modular relation for the chromatic symmetric functions of (3+1)-free posets