  We study the joint distribution of descents and inverse descents over the set
of permutations of n letters. Gessel conjectured that the two-variable
generating function of this distribution can be expanded in a given basis with
nonnegative integer coefficients. We investigate the action of the Eulerian
operators that give the recurrence for these generating functions. As a result
we devise a recurrence for the coefficients but are unable to settle the
conjecture. We examine generalizations of the conjecture and obtain a type B
analog of the recurrence satisfied by the two-variable generating function. We
also exhibit some connections to cyclic descents and cyclic inverse descents.
Finally, we propose a combinatorial model in terms of statistics on inversion
sequences.

Combinatorics