On Whitney numbers of the first and second kind, or is it the other way around?

Dr. Rafael S. González D’León
Assistant Professor 
Loyola University Chicago

Algebraic combinatorics is centered on the idea that a variety of algebraic objects can be understood using the tools from combinatorics. In this talk, I will illustrate this idea using a modern example of how combinatorial theories can give us a better understanding on a subject. The story I will tell involves the Whitney numbers of the first and second kind, which are a pair of invariants of partially ordered sets (posets) that are relevant in several areas of mathematics. One of their most interesting appearances is as the coefficients of a polynomial enumerating proper colorings of a graph (think of colorings of a map where neighboring countries must be distinguishable from each other). They also appear counting regions in the complement of a real hyperplane arrangement.

I will introduce and give insight into the algebraic and combinatorial objects in this story and share a very curious phenomenon: sometimes the Whitney numbers of the first and second kind of a poset are also the Whitney numbers of the second and first kind, but of a different poset. Some recent results about this phenomenon could shed light on longstanding questions in the subject.