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A conjectured combinatorial formula for the Hilbert series for diagonal harmonics. (English) Zbl 1070.05007

Summary: 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 J. Haglund and A. M. Garsia [Proc. Natl. Acad. Sci. USA 98, 4313-4316 (2001; Zbl 1066.05144)]. 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 sequence introduced by Haglund and the other of which is expressed in terms of rooted, labelled trees.

MSC:

05A15 Exact enumeration problems, generating functions
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
05C05 Trees

Citations:

Zbl 1066.05144
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References:

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