J. Haglund

A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants (condensed version)
dmtcs:2924 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)

https://doi.org/10.46298/dmtcs.2924
A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants (condensed version)
Authors: J. Haglund
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J. Haglund
A special case of Haiman's identity [Invent. Math. 149 (2002), pp. 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in $q,t$. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman's formula for the Hilbert series into an explicit polynomial in $q,t$ with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.