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Báo cáo toán học: " THE DISTRIBUTION OF DESCENTS AND LENGTH IN A COXETER GROUP"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: THE DISTRIBUTION OF DESCENTS AND LENGTH IN A COXETER GROUP. | THE DISTRIBUTION OF DESCENTS AND LENGTH IN A COXETER GROUP Victor Reiner University of Minnesota e-mail reiner@math.umn.edu Submitted August 19 1995 accepted November 25 1995 Abstract. We give a method for computing the q-Eulerian distribution W t q 2 tdes w ql w w2W as a rational function in t and q where W S is an arbitrary Coxeter system l w is the length function in W and des w is the number of simple reflections s 2 S for which l ws l w . Using this we compute generating functions encompassing the q-Eulerian distributions of the classical infinite families of finite and affine Weyl groups. I. Introduction. Let W S be a Coxeter system see Hu for definitions and terminology . There are two statistics on elements of the Coxeter group W l w min l w si1 si2 Sit for some sik 2 Sg des w s 2 S l ws l w g which generalize the well-known permutation statistics inversion number and descent number in the case W is the symmetric group Sn. The polynomial X t ies w w2Sn is known in the combinatorial literature as the Eulerian polynomial which has generating function X xl X tdes w 1 t ex 1 tl n 1 t è ỵ 1 n 0 w2Sn 1991 Mathematics Subject Classification. 05A15 33C80. Work supported by Mathematical Sciences Postdoctoral Research Fellowship DMS-9206371 Typeset by Ạuí S-TeX THE ELECTRONIC JOURNAL OF COMBINATORICS 2 1995 R25 2 and a q-analogue first computed by Stanley St 3 1 X xn X tdes w qi w 1 - t exp x 1 - t q 0 n 41 1 - t exp x 1 - t q b 0 w 2 Sn where exp x q is the q-exponential given by X xn exp x q Y n 0 n n n x q n using the notation n n - 1 2 1 1 ỳ 1 - qn 1 - q 1 - x 1 - qx 1 - q2x 1 - qn 1x For this reason we call W t q tdes w ql w w2W the q-Eulerian distribution of the Coxeter system W S or the q-Eulerian distribution of W by abuse of notation. We caution the reader that this is not the same notion as the q-Eulerian polynomial considered in Br for W Bn Dn . Analogous generating functions to equation 1 for the infinite families of finite Coxeter groups W Bn Cn Dn were