A conjectured combinatorial formula for the Hilbert series for diagonal harmonics

We introduce a conjectured way of expressing the Hilbert series of diagonal harmonics as a weighted sum over parking functions. Our conjecture is based on a pair of statistics for the q,t-Catalan sequence discovered by Haiman and proven by Haglund and Garsia (Proc. Nat. Acad. Sci. 98 (2001) 4313-4316). We show how our q,t-parking function formula for the Hilbert series can be expressed more compactly as a sum over permutations. We also derive two equivalent forms of our conjecture, one of which is based on the original pair of statistics for the q,t-Catalan introduced by Haglund and the other of which is expressed in terms of rooted, labelled trees.

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