Many of the classic counting problems in combinatorics -- like counting subsets of a set, colorings of a graph, or integer points in a polytope -- can be answered with polynomial formulas. As these are polynomials, we can try plugging negative numbers into them, but it's not immediately clear what this means. What are colorings of a graph with -1 colors, or subsets of a set of -5 elements, or integer points in a cube scaled by a factor of -2?

In the first part of this talk, we'll go through the suspiciously similar answers to these questions. In the second part, we'll state an elegant theorem of Stanley that generalizes all of them using generating functions, and unpack why it is a generalization. Speaker(s): Will Dana