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: Unimodal rays in the regular and generalized Pascal pyramids. | Unimodal rays in the regular and generalized Pascal pyramids Hacene Belbachir USTHB Faculty of Mathematics Po. Box 32 El Alia Algiers Algeria hbelbachir@ László Szalayi University of West Hungary Institute of Mathematics and Statistics H-9400 Sopron Erzsebet utca 9. Hungary laszalay@ Submitted Jun 15 2009 Accepted Mar 25 2011 Published Mar 31 2011 Mathematics Subject Classification 11B65 11B83 Abstract The present paper shows that any sequence of integers laying along a finite ray in any Pascal-pyramid is log-concave consequently unimodal. Further we describe an algorithm which provides the plateaus of the diagonal sequences locating on layer n of the regular Pascal pyramid. 1 Introduction Let r 2 denote an integer and consider the map p Nr N . i X1 xr xi xr X1 . . . Xr ---- ------- X1 . xr x1 X r Clearly the map p provides the number of the ways of splitting a set of x x1 xr distinguishable objects into pairwisely disjoint subsets Si with cardinality x i 1 . r therefore pT1 ----- xr x1 . . . xr x1 xr x1 Kx2 xr x2 xr -1 xr xr-1 xxrr 1 The research is supported by LAID3 Laboratory of USTHB University and by Algerian-French bilateral CMEP-Tassili 09MDU765. The research is supported by Janos Bolyai Scholarship of HAS by Hungarian National Foundation for Scientific Research Grant No. T 61800 FT and by Mexican-Hungarian bilateral TeT. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P79 1 holds with the usual binomial coefficients on the right hand side. When r 2 the map returns the regular binomial coefficients in the Pascal triangle. We obtain the regular Pascal pyramid if r 3. For the cases r 4 we will refer as generalized Pascal pyramids. The main purpose of this paper is to investigate the question of unimodality in Pascal pyramids. It has already been shown that the elements laying along any ray in the Pascal triangle form unimodal sequence see 2 . Scrutinies were made to reveal the localization of certain rays modes for instance see 5 3 1 . Here
