  Let $I$ be the ideal generated by alternating polynomials in two sets of $n$
variables. Haiman proved that the $q,t$-Catalan number is the Hilbert series of
the graded vector space $M(=\bigoplus_{d_1,d_2}M_{d_1,d_2})$ spanned by a
minimal set of generators for $I$. In this paper we give simple upper bounds on
$\text{dim}M_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees
$(d_1,d_2)$ such that $\dim M_{d_1, d_2}$ achieve the upper bounds. For such
bi-degrees, we also find explicit bases for $M_{d_1, d_2}$. The main idea is to
define and study a nontrivial linear map from $M$ to a polynomial ring
$\C[\rho_1, \rho_2,...]$.
