The bigraded Frobenius characteristic of the Garsia-Haiman module $\BM_\mu$ is known to be given by the Macdonald polynomial $\TH_\mu[X;q,t]$. The bigraded Hilbert series $F_\mu(q,t)$ of $\BM_\mu$ may be calculated from a recursion of the form $ F_\mu (q,t) = \sum_{\nu\rightarrow \mu}c_{\mu\nu}F_\nu (q,t)$. where $\nu\rightarrow \mu$ means $\nu$ is an immediate predecessor of $\mu$ in the lattice of partitions. The theory of Macdonald polynomials gives explicit expressions for the coefficients $c_{\mu\nu }$ as rational fumctions of $q,t$. One of the challenging problems of the theory of Macdonald polynomials has been to explain why, in spite of the forbidding intricacy of the $c_{\mu\nu }$, the $F_\mu (q,t)$ turn out to be such beautiful polynomials. We give here a new recursion, from which a new combinatorial formula for $F_\mu (q,t)$ can be derived when $\mu$ is a two column partition. The proof suggests a method for deriving an analogous formula in the general case.

Pre-seminar: Wanted: n! derivatives 1000$ reward.
We give here a combinatorial recipe for proving the so called ``n! conjecture'' which is now a Theorem proved by M. Haiman by algebraic geometrical methods. It is shown how some simple geometrical properties of trees can yield surprising results in analysis and algebra.