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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: An analogue of the Thue-Morse sequence. | An analogue of the Thue-Morse sequence Emmanuel FERRAND Institut Mathématique de Jussieu UMR CNRS 7586 Universite Pierre et Marie Curie Paris VI Case 247 - 4 place Jussieu - 75252 Paris Cedex FRANCE ferrand@math.jussieu.fr Submitted Sep 21 2005 Accepted Feb 23 2007 Published Apr 23 2007 Mathematics Subject Classification 05C88 Abstract We consider the finite binary words Z n n 2 N defined by the following selfsimilar process Z 0 0 Z 1 01 and Z n 1 Z n Z n 1 where the dot denotes word concatenation and w the word obtained from w by exchanging the zeros and the ones. Denote by Z 1 01110100. the limiting word of this process and by z n the n th bit of this word. This sequence z is an analogue of the Thue-Morse sequence. We show that a theorem of Bacher and Chapman relating the latter to a Sierpmski matrix has a natural analogue involving z. The semi-infinite self-similar matrix which plays the role of the Sierpinski matrix here is the zeta matrix of the poset of finite subsets of N without two consecutive elements ordered by inclusion. We observe that this zeta matrix is nothing but the exponential of the incidence matrix of the Hasse diagram of this poset. We prove that the corresponding Mobius matrix has a simple expression in terms of the zeta matrix and the sequence z . 1 Introduction Consider the finite binary words T n n 2 N defined by the following self-similar process T 0 0 and T n 1 T n T n where the dot denotes word concatenation and w the word obtained from w by exchanging the zeros and the ones. Denote by T 1 01101001. the limiting word of this process and by t n the n th bit of this word. The sequence t is often called the Thue-Morse sequence and has appeared in various fields of mathematics. See for example the paper AS1 which contains a review of the main properties of this sequence and which is a good starting point to the abundant literature on the subject1. -LSee also AS2 6.2 THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R30 1 In BC Bacher and .
