Geodesic
Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonic modules
The electronic journal of combinatorics, Tome 11 (2004) no. 1
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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 Hilbert series of certain generalizations of the diagonal harmonics modules.
DOI : 10.37236/1821
Classification : 05E05, 05A10, 11B65, 20C30
Mots-clés : Dyck paths, Macdonald polynomials, Nabla operator, statistics 
@article{10_37236_1821,
     author = {Nicholas A. Loehr and Jeffrey B. Remmel},
     title = {Conjectured combinatorial models for the {Hilbert} series of generalized diagonal harmonic modules},
     journal = {The electronic journal of combinatorics},
     year = {2004},
     volume = {11},
     number = {1},
     doi = {10.37236/1821},
     zbl = {1062.05141},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1821/}
}
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Nicholas A. Loehr; Jeffrey B. Remmel. Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonic modules. The electronic journal of combinatorics, Tome 11 (2004) no. 1. doi: 10.37236/1821

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