Doppelgangers: the Ur-Operation and Posets of Bounded Height

TL;DR

This paper characterizes when different posets have the same order polynomial, introduces a new poset operation called the Ur-operation, and provides efficient algorithms for classifying doppelgangers and computing order polynomials of bounded height posets.

Contribution

It introduces the Ur-operation for posets, characterizes doppelgangers, and develops algorithms for classifying and computing order polynomials of bounded height posets.

Findings

Doppelgangers can be classified via systems of diophantine equations in polynomial time.

Order polynomial computation for bounded height posets is linear in the size of the poset.

Necessary and sufficient conditions for poset doppelgangers are established using new and existing tools.

Abstract

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.