On Whitney numbers of the first and second kind, or is it the other way around?
Rafael S. González D'León
3 Nov 2023
The Whitney numbers of the first and second kind are a pair of poset invariants that are relevant in various areas of mathematics. One of the most interesting appearances of these numbers is as the coefficients of the chromatic polynomial of a graph. They also appear as counting regions in the complement of a real hyperplane arrangement. In this talk, I will share a very curious phenomenon: sometimes the Whitney numbers of the first and second kind of a poset happen to be also the Whitney numbers of the second and first kind but of a different poset. We call this phenomenon Whitney duality and to find examples we rely on the techniques of edge labelings and quotient posets. I will present some key results in the theory of Whitney duality and in particular recent results regarding nonuniqueness. Joint work with Josh Hallam and Yeison Quiceno.