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We prove a combinatorial formula for the Hilbert series of the Garsia-Haiman bigraded $S_n$-modules as weighted sums over standard Young tableaux in the hook shape case. This method is based on the combinatorial formula of Haglund, Haiman and Loehr for the Macdonald polynomials and extends the result of A. Garsia and C. Procesi for the Hilbert series when $q=0$. Moreover, we construct an association of the fillings giving the monomial terms of Macdonald polynomials with the standard Young tableaux.

A Combinatorial Formula for the Hilbert Series of Bigraded $S_n$-Modules