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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On unimodality problems in Pascal’s triangle. | On unimodality problems in Pascal s triangle Xun-Tuan Su and Yi Wang y Department of Applied Mathematics Dalian University of Technology Dalian 116024 P. R. China suxuntuan@yahoo.com.cn wangyi@dlut.edu.cn Submitted Jan 23 2008 Accepted Aug 28 2008 Published Sep 8 2008 Mathematics Subject Classification 05A10 05A20 Abstract Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We also propose two more general conjectures. 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 such an integer m is called a mode of the sequence . In particular a monotone increasing or decreasing sequence is known as unimodal. The sequence is called concave resp. convex if for i 1 ai-1 ai 1 2ai resp. ai_1 ai 1 2ai . The sequence is called log-concave resp. log-convex if for all i 1 ai_1ai 1 a2 resp. ai_1 ai 1 a2 . By the arithmetic-geometric mean inequality the concavity implies the log-concavity the log-convexity implies the convexity . For a sequence ai of positive numbers it is log-concave resp. log-convex if and only if the sequence ai 1 ai is decreasing resp. increasing and so the log-concavity implies the unimodality. The unimodality problems including concavity convexity and logconcavity log-convexity arise naturally in many branches of mathematics. For details see 3 4 13 17 18 19 21 22 about the unimodality and log-concavity and 7 10 about the log-convexity. Partially supported by the National Science Foundation of China under Grant No.10771027. y Corresponding author. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R113 1 Many sequences of binomial coefficients share various .
