Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules. | Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules Nicholas A. Loehr Department of Mathematics University of Pennsylvania Philadelphia PA 19104 nloehr@ Jeffrey B. Remmel Department of Mathematics University of California at San Diego La Jolla CA 92093 jremmel@ Submitted Jul 31 2003 Accepted Sep 5 2004 Published Sep 24 2004 Mathematics Subject Classifications 05A10 05E05 05E10 20C30 11B65 Abstract 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. 1 Introduction A Dyck path of order n is a path in the x -plane from 0 0 to n n consisting of n vertical steps and n horizontal steps each of length one such that no step goes strictly Supported by a National Science Foundation Graduate Research Fellowship THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R68 1 below the diagonal line y x. A labelled Dyck path is a Dyck path whose vertical steps are labelled 1 2 . n in such a way that the labels for .
