Title: The Poset Conjecture: results, counterexamples, and open problems
|University of Waterloo
|Contact Karen Yeats
In 1978, Neggers conjectured that a certain transform of the order polynomial of a partially ordered set (poset) has only real roots.
In the late 1980s, Stanley gave this to me as a thesis project, generalized to labelled posets. For my thesis I proved the conclusion for series-parallel labelled posets and a bit more. Br\"and\'en, and later Stembridge, found counterexamples to the conjecture in general.
But the conjecture remains open for some notable subclasses of the class of all posets. Ferrers posets -- unknown -- clearly important.
Posets for which the Hasse (covering) graph is a tree might be possible. The case in which the Hasse graph is a path is still open, but seems almost within reach given some new ideas.