51ÁÔÆæ

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

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.