  In the early 1970's, Richard Stanley and Kenneth Johnson introduced and laid
the groundwork for studying the order polynomial of partially ordered sets
(posets). Decades later, Hamaker, Patrias, Pechenik, and Williams introduced
the term "doppelgangers": equivalence classes of posets under the equivalence
relation given by equality of the order polynomial. We provide necessary and
sufficient conditions on doppelgangers through application of both old and
novel tools, including new recurrences and the Ur-operation: a new generalized
poset operation. In addition, we prove that the doppelgangers of posets P of
bounded height $|P|-k$ may be classified up to systems of $k$ diophantine
equations in $2^{O(k^2)}$ time, and similarly that the order polynomial of such
posets may be computed in $O(|P|)$ time.
