An Analogue of the Thue-Morse Sequence

We consider the finite binary words Z(n), n ∈ 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(∞) = 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 “Sierpi´nski matrix” has a natural analogue involving z. The semi-infinite self-similar matrix which plays the role of the Sierpi´nski 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 M¨obius matrix has a simple expression in terms of the zeta matri...

• Type: Journal
• Year: 2007
• Where: COMBINATORICS
• Authors: Emmanuel Ferrand
• Keywords: COMBINATORICS 2007, Denotes Word Concatenation, Matrix, finite Binary Words