Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonic modules
In a previous paper, Haglund and Loehr conjectured two equivalent combinatorial formulas for the Hilbert series of the Garsia-Haiman diagonal harmonic modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. In the paper under review, the authors introduce a third combinatorial formula that is shown to be equivalent to the previous two. The authors also show that the four statistics on the labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles a question asked earlier by Haglund and Loehr. Analogous statistics are then introduced on other collections of labelled lattice paths contained in trapezoids, and a fermionic formula is obtained for their generating function. The authors give bijective proofs of the equivalence of several forms of this generating function, which imply that all the new statistics have the same univariate distribution. Finally, using these new statistics, several combinatorial formulas are conjectured for the Hilbert series of certain generalizations of the diagonal harmonic modules.
scientific article; zbMATH DE number 2161298
scientific article