On a decomposition of the two-sided Eulerian polynomials

A conjecture of Ira Gessel asserts that the two-sided Eulerian polynomial - the joint generating polynomial of descents and inverse descents admits a decomposition where the coefficients of are nonnegative integers. In this talk, we derive a recurrence satisfied by these coefficients, which brings us a step closer to resolving the conjecture. We also mention further possible lines of attack, including a multivariate generalization of a classical theorem of Foata-Schutzenberger and describe a conjectured combinatorial model for the two-sided Eulerian polynomials in terms of inversion sequences.