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Báo cáo toán học: "A recurrence relation for the “inv” analogue of q-Eulerian polynomials"

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: A recurrence relation for the “inv” analogue of q-Eulerian polynomials. | A recurrence relation for the inv analogue of q-Eulerian polynomials Chak-On Chow Department of Mathematics and Information Technology Hong Kong Institute of Education 10 Lo Ping Road Tai Po New Territories Hong Kong cchow@alum.mit.edu Submitted Feb 23 2010 Accepted Apr 12 2010 Published Apr 19 2010 Mathematics Subject Classifications 05A05 05A15 Abstract We study in the present work a recurrence relation which has long been overlooked for the q-Eulerian polynomial Anes inv t q Eheeg des ơ qinv ơ where des ơ and inv ơ denote respectively the descent number and inversion number of Ơ in the symmetric group n of degree n. We give an algebraic proof and a combinatorial proof of the recurrence relation. 1 Introduction Let Sn denote the symmetric group of degree n. Any element ơ of Sn is represented by the word ơ1 ơ2 ơn where ơi ơ i for i 1 2 . n. Two well-studied statistics on n are the descent number and the inversion number defined by n des ơ 22 x ơi Ơi 1 i 1 inv ơ 22 x ơi ơ 1Cí jCn respectively where ơn 1 0 and x P 1 or 0 depending on whether the statement P is true or not. It is well-known that des is Eulerian and that inv is Mahonian. The generating function of the Euler-Mahonian pair des inv over Sn is the following q- Eulerian p olynomial anes inv t q tdes 0qinvC . ơeSn THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N22 1 It is clear that An t 1 An t the classical Eulerian polynomial. Let z and q be commuting indeterminates. For n 0 let n q 1 q q2 qn-1 be a q-integer and n q 1 q 2 q n q be a q-factorial. Define a q-exponential function by zn q Lt l n 0 n q Stanley 6 proved that xn 1 t Ades inv x-t q g An Jnv t.q 1 - te x-1 - t q 1 Alternate proofs of 1 have also been given by Garsia 4 and Gessel 5 . Desarmenien and Foata 2 observed that the right side of 1 is precisely f1 t V 1 nn-1 x V. V - ằí 1 -t w and from which they obtained a semi q-recurrence relation for Anes inv t q namely n i L -I q t q t 1 - t n-1 1 i n-1 Ades inv t q t 1 - t n-1-i. The above .