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Báo cáo toán học: "Conjectured Statistics for the Higher q, t-Catalan Sequences"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Conjectured Statistics for the Higher q, t-Catalan Sequences. | Conjectured Statistics for the Higher q -Catalan Sequences Nicholas A. Loehr Department of Mathematics University of Pennsylvania Philadelphia PA 19104 nloehr@math.upenn.edu Submitted Oct 22 2002 Accepted Jan 24 2005 Published Feb 14 2005 Mathematics Subject Classifications 05A10 05E05 05E10 20C30 11B65 Abstract This article describes conjectured combinatorial interpretations for the higher q t-Catalan sequences introduced by Garsia and Haiman which arise in the theory of symmetric functions and Macdonald polynomials. We define new combinatorial statistics generalizing those proposed by Haglund and Haiman for the original q t-Catalan sequence. We prove explicit summation formulas bijections and recursions involving the new statistics. We show that specializations of the combinatorial sequences obtained by setting t 1 or q 1 or t 1 q agree with the corresponding specializations of the Garsia-Haiman sequences. A third statistic occurs naturally in the combinatorial setting leading to the introduction of q t r-Catalan sequences. Similar combinatorial results are proved for these trivariate sequences. 1 Introduction In 7 Garsia and Haiman introduced a q -analogue of the Catalan numbers which they called the q -Catalan sequence. In the same paper they introduced a whole family of higher q -Catalan sequences one for each positive integer m. We begin by describing several equivalent characterizations of the original q -Catalan sequence. We then discuss analogous characterizations of the higher q -Catalan sequences. In the rest of the paper we present some conjectured combinatorial interpretations for the higher q t-Catalan sequences. We prove some combinatorial formulas recursions and Work supported by an NSF Graduate Research Fellowship and an NSF Postdoctoral Research Fellowship THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R9 1 bijections and introduce a three-variable version of the Catalan sequences. We also show that certain specializations of our combinatorial .