tailieunhanh - Báo cáo toán học: " Unimodality problems of multinomial coefficients and symmetric func Unimodality problems of multinomial coefficients and symmetric functiontion"

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: Unimodality problems of multinomial coefficients and symmetric functions. | Unimodality problems of multinomial coefficients and symmetric functions Xun-Tuan Su1 Yi Wang2 1 2School of Mathematical Sciences Dalian University of Technology Dalian 116024 P. R. China 1suxuntuan@ 2wangyi@ Yeong-Nan Yeh3 t 3Institute of Mathematics Academia Sinica Taipei 10617 Taiwan 3mayeh@ Submitted Nov 14 2010 Accepted Mar 20 2011 Published Mar 31 2011 Mathematics Subject Classification 05A10 05A20 Abstract In this note we consider unimodality problems of sequences of multinomial coefficients and symmetric functions. The results presented here generalize our early results for binomial coefficients. We also give an answer to a question of Sagan about strong q-log-concavity of certain sequences of symmetric functions which can unify many known results for q-binomial coefficients and q-Stirling numbers of two kinds. Keywords Unimodality Log-concavity Log-convexity q-log-concavity Strong q-log-concavity Multinomial coefficients Symmetric functions 1 Introduction Let a0 a1 a2 . be a sequence of nonnegative numbers. It is called unimodal if a0 a1 am-1 am am 1 for some m. It is called log-concave resp. log-convex if ai_1ai 1 a2 resp. ai_1ai 1 a2 for all i 1. Clearly a sequence ai of positive numbers is log-concave resp. log-convex if and only if ai_1aj 1 aiaj resp. ai_1aj 1 aiaj for 1 i j. So the log-concavity of a sequence of positive numbers implies the unimodality. Corresponding author. Partially supported by the National Science Foundation of China. Partially supported by NSC 98-2115-M-001-010. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P73 1 Unimodality problems including unimodality log-concavity and log-convexity of sequences arise naturally in combinatorics and other branches of mathematics see . 1 2 6 7 9 12 14 15 . In particular many sequences of binomial coefficients enjoy various unimodality properties. For example the sequence of binomial coefficients along any finite transversal of Pascal s triangle is .