The Hilbert series of the Garsia-Haiman module can be defined combinatorially as generating functions of certain fillings of Ferrers diagrams. One of the challenges in working with the combinatorial definition is the large number of fillings needed to generate a polynomial. In this talk we look at combinatorial proof of some factorizations of the Hilbert series.

The cd-index of an Eulerian poset is a polynomial in noncommuting variables that is used to encode data on the number of chains within that poset.  I will focus in this talk on a particular class of Eulerian posets involving the Bruhat order of Coxeter groups.  I will use poset operations such as zipping and the pyramid operation to develop Reading's recursive formula for the cd-index of Bruhat intervals.

There are many combinatorial interpretations of Stirling numbers of the first and second kinds. In this talk I will focus on the interpretations of de Medicis and Leroux involving 0-1 tableaux and 0-1 matrices. I will describe how they use them to prove identities involving Stirling numbers, such as orthogonality and Carlitz' identity.

A problem of recent importance has been to identify when a lattice polytope has a symmetric and unimodular h*-vector. In the 1960s, Stanley showed that symmetry occurs exactly when the polytope is Gorenstein. In 2003, Athanasiadis gave sufficient conditions for unimodality.  In this talk, we will discuss a result by Bruns and Römer that generalizes these conditions.