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@ 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 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 .

Không thể tạo bản xem trước, hãy bấm tải xuống
TÀI LIỆU MỚI ĐĂNG
12    30    2    05-07-2022
Đã phát hiện trình chặn quảng cáo AdBlock
Trang web này phụ thuộc vào doanh thu từ số lần hiển thị quảng cáo để tồn tại. Vui lòng tắt trình chặn quảng cáo của bạn hoặc tạm dừng tính năng chặn quảng cáo cho trang web này.