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COMBINATORICS
2004

Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules

8 years 5 months ago
Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules
Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the Garsia-Haiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hi...
Nicholas A. Loehr, Jeffrey B. Remmel
Added 17 Dec 2010
Updated 17 Dec 2010
Type Journal
Year 2004
Where COMBINATORICS
Authors Nicholas A. Loehr, Jeffrey B. Remmel
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