MIT Combinatorics Seminar

Topology of Rees products of posets and q-Eulerian polynomials

Michelle Wachs  (University of Miami)

Wednesday, March 14, 2007    4:15 pm    Room 2-136


A poset operation called Rees product was recently introduced by Björner and Welker in their study of connections between poset topology and commutative algebra. Through our study of Rees products of posets, we discovered a remarkable q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. Although poset topology led us to conjecture our formula, symmetric function theory provided the proof. We prove a symmetric function generalization of our formula, which yields other enumerative results. In our work we also establish a connection between the cohomology of the toric variety associated with the Coxeter complex of type A (studied by Procesi, Stanley, Stembridge, and Dolgachev and Lunts), and the homology of certain intervals in the Rees product of a (partially truncated) Boolean algebra and a chain, as modules for the symmetric group.

This is joint work with John Shareshian.