Zeros of Eulerian polynomials and Permanents connected to Rook Theory

Polynomials with only real zeros have become an important subject in algebraic combinatorics. A classic example of this is the Eulerian polynomials, which arose in Euler's work on series but also enumerate permutations by number of descents. In this talk we discuss various generalizations of the fact that the Eulerian polynomials have only real zeros. One of these is a result of Branden, Visontai, Wagner and the speaker, the so-called Monotone Column Permanent Theorem, which grew out of the study of rook polynomials.