A combinatorial formula for the character of the diagonal convariants

Let R-n be the ring of coinvariants for the diagonal action of the symmetric group S-n. It is known that the character of R-n as a doubly graded S-n-module can be expressed using the Frobenius characteristic map as del e(n), where e(n) is the nth elementary symmetric function and del is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for del e(n) and prove that it has many desirable properties that support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc, and Thibon. We also show that a variety, of earlier conjectures and theorems on del e(n) are special cases of our conjecture. Finally, we extend our conjectures on del e(n) and several on the results supporting them to higher powers del(m)e(n).