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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Rook Theory, Generalized Stirling Numbers and (p, q)-analogues. | Rook Theory Generalized Stirling Numbers and P ợ -analogues J. B. Remmel Department of Mathematics University of California at San Diego La Jolla CA 92093-0112 jremmel@math.ucsd.edu Michelle L. Wachs Department of Mathematics University of Miami Coral Gables FL 33124 wachs@math.miami.edu Submitted Jun 28 2004 Accepted Oct 21 2004 Published Nov 22 2004 MR Subject Classification 05A05 05A18 05A19 05A30 Abstract In this paper we define two natural p q -analogues of the generalized Stirling numbers of the first and second kind S 1 a 3 r and S2 a 3 r as introduced by Hsu and Shiue 17 . We show that in the case where 3 0 and a and r are nonnegative integers both of our p q -analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our p q -analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our p q -analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics. 1 Introduction In this paper we present a new rook theory interpretation of a certain class of generalized Stirling numbers and their p q -analogues. Our starting point is to develop two natural p q -analogues of the generalized Stirling numbers as defined by Hsu and Shiue in 17 . That is Hsu and Shiue gave a unified approach to many extensions of the Stirling numbers that had appeared in the literature by defining analogues of the Stirling numbers of the first and second kind which depend on three parameters a 3 and r as follows. First define z a 0 1 and z a ra z z a z n 1 a for each integer n 0. We write z ịn for z a ra when a 1 Supported in part by NSF grant DMS 0400507 1 Supported in part by NSF grant DMS 0300483 THE .
