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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: On the Modes of Polynomials Derived from Nondecreasing Sequences. | On the Modes of Polynomials Derived from Nondecreasing Sequences Donna Q. J. Dou School of Mathematics Jilin University Changchun 130012 P. R. China qj dou@jlu.edu.cn Arthur L. B. Yang Center for Combinatorics LPMC-TJKLC Nankai University Tianjin 300071 P. R. China yang@nankai.edu.cn Submitted Oct 13 2010 Accepted Dec 15 2010 Published Jan 5 2011 Mathematics Subject Classification 05A20 33F10 Abstract Wang and Yeh proved that if P x is a polynomial with nonnegative and nondecreasing coefficients then P x d is unimodal for any d 0. A mode of a unimodal polynomial f x a0 a1x amxm is an index k such that ak is the maximum coefficient. Suppose that M P d is the smallest mode of P x d and M P d the greatest mode. Wang and Yeh conjectured that if d2 di 0 then M P d1 M P d2 and M P d1 M P d2 . We give a proof of this conjecture. Keywords unimodal polynomials the smallest mode the greatest mode. 1 Introduction This paper is concerned with the modes of unimodal polynomials constructed from nonnegative and nondecreasing sequences. Recall that a sequence a-. o - m is unimodal if there exists an index 0 k m such that a0 ak-1 ak ak 1 am- Such an index k is called a mode of the sequence. Note that a mode of a sequence may not be unique. The sequence ai 0 i m is said to be spiral if am ao Om-1 ai a m 1.1 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P1 1 where m stands for the largest integer not exceeding 22. Clearly the spiral property implies unimodality. We say that a sequence ai 0 i m is log-concave if for 1 k m 1 2 n .n ak ak 1ak-1 and it is ratio monotone if am a0 Zm- a1 ai am- m-1 a 1 1.2 and a0 am- a1 1 am-2 ai-1 am-i a m -1 - am- m 1. 1.3 It is easily checked that ratio monotonicity implies both log-concavity and the spiral property. Let P x a0 a1x amxm be a polynomial with nonnegative coefficients. We say that P x is unimodal if the sequence ai 0 i m is unimodal. A mode of ai 0 i m is also called a mode of P x . Similarly we say that P x is log-concave or ratio .
