tailieunhanh - Báo cáo toán học: "The Generalized Schr¨der Theory o"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The Generalized Schr¨der Theory o. | The Generalized Schroder Theory Chunwei Song Dept of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama 2-12-1 Meguro-ku Tokyo 152-8552 Japan csong@ Submitted Apr 18 2005 Accepted Oct 12 2005 Published Oct 20 2005 Mathematics Subject Classification 05A30 05C30 05A99 Abstract While the standard Catalan and Schroder theories both have been extensively studied people have only begun to investigate higher dimensional versions of the Catalan number see say the 1991 paper of Hilton and Pedersen and the 1996 paper of Garsia and Haiman . In this paper we study a yet more general case the higher dimensional Schroder theory. We define m-Schroder paths find the number of such paths from 0 0 to mn n and obtain some other results on the m-Schroder paths and m-Schroder words. Hoping to generalize classical q-analogue results of the ordinary Catalan and Schroder numbers such as in the works of Furlinger and Hofbauer Cigler and Bonin Shapiro and Simion we derive a q-identity which would welcome a combinatorial interpretation. Finally we introduce the q t -m-Schroder polynomial through m-parking functions and relate it to the m-Shuffle Conjecture of Haglund Haiman Loehr Remmel and Ulyanov. 1 Introduction Throughout this paper we use the standard notation M l - qn 1 - q . M- l 2 . n n yy for the q-analogue of the integer n the q-factorial and the q-binomial coefficient and a n 1 a 1 qa 1 qn-1a for the q-rising factorial. Sometimes it is necessary to write the base q explicitly as in n q. n q n q and a q n but we omit q in this paper since it is clear from the context. When i j k n . . ijiijjqfe represents the q-trinomial coefficient. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R53 1 Definition A Dyck path of order n is a lattice path from 0 0 to n n that never goes below the main diagonal i i 0 i ng with steps 0 1 or NORTH for brevity N and 1 0 or EAST for brevity E . Let Dn denote the set of all Dyck paths of order n. An example of a .