Counting objects up to symmetry is one of the fundamental problems of enumerative combinatorics. In this thesis, we study this problem in the context of partially ordered sets. We first present the basics of group actions on sets and prove Burnside's lemma and Pólya's enumeration theorem. We then introduce Stanley's order polynomial, which counts order-preserving maps from a partially ordered set to a chain, express it in terms of the descents of linear extensions, and prove the reciprocity theorem. Finally, we combine these two theories and introduce the orbital order polynomial, which counts order-preserving maps up to symmetry. We show that Pólya's enumeration theorem arises as a special case and prove the corresponding reciprocity theorem.
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