Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: q, t-Catalan numbers and generators for the radical ideal defining the diagonal locus of (C2)n. | q t-Catalan numbers and generators for the radical ideal defining the diagonal locus of C2 n Kyungyong Lee Department of Mathematics University of Connecticut Storrs CT 06269 . Li Li Department of Mathematics and Statistics Oakland University Rochester MI 48309 . li2345@ Submitted Dec 6 2010 Accepted Jul 28 2011 Published Aug 5 2011 Mathematics Subject Classifications 05E15 05E40 Abstract Let I be the ideal generated by alternating polynomials in two sets of n variables. Haiman proved that the q t-Catalan number is the Hilbert series of the bi-graded vector space M 0d1 d2 Md1 d2 spanned by a minimal set of generators for I. In this paper we give simple upper bounds on dim Md1 d2 in terms of number of partitions and find all bi-degrees d1 d2 such that dimMd1 d2 achieve the upper bounds. For such bi-degrees we also find explicit bases for Md1 d2. 1 Introduction In 6 Garsia and Haiman introduced the q t-Catalan number Cn q t and showed that Cn q 1 agrees with the q-Catalan number defined by Carlitz and Riordan 3 . To be more precise take the n X n square whose southwest corner is 0 0 and northeast corner is n n . Let Dn be the collection of Dyck paths . lattice paths from 0 0 to n n that proceed by NORTH or EAST steps and never go below the diagonal. For any Dyck path n define area n to be the number of lattice squares below n and strictly above the diagonal. Then C q 1 V qarea n . The q t-Catalan number Cn q t also has a combinatorial interpretation using Dyck paths. Given a Dyck path n let ai n be the number of squares in the i-th row that lie in the region bounded by n and the diagonal and define dinv n i j i j and ữựn aj n I i j i j and ữựn 1 aj n I. Partially supported by NSF grant DMS 0901367 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P158 1 In 4 1 and 5 Theorem Garsia and Haglund showed the following combinatorial formula 1 Cn q t 2 qarea n tdinv n . A natural question is to find the coefficient of .
