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.