A sort-of heuristic in combinatorics is that if you can't figure out what to do with a set, take the free abelian group / vector space on that set and work with linear transformations instead of functions. For example, there is a certain condition on certain linear operators one can define on the free vector space on a graded poset that guarantees the ranks are unimodal, and for many applications of this condition a direct proof of unimodality isn't known.
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