A conjectured combinatorial formula for the Hilbert series for diagonal harmonics

Abstract

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.