The complexity of Cbw-words of the form w×w

• Authors: Yunbao Huang
• Venue: Theoretical Computer Science
• Year: 2009

Let p"n(~) denote the number of C^b^@w-words of the form w@?xw with gap n and p"n(k) denote the number of C^~-words of the form w@?xw with length 2k+n and gap n, where n is the length of the word x. [S. Brlek, A. Ladouceur, A note on differentiable palindromes, Theoret. Comput. Sci. 302 (2003) 167-178] proved that C^~-palindromes are characterized by the left palindromic closure of the prefixes of the well-known Kolakoski sequences and revealed an interesting perspective for understanding some of the conjectures. In fact, they found all infinite C^~-palindromes and established p"0(k)=p"1(k)=2 for all k@?N, where N is the set of positive integers. [Y.B. Huang, About the number of C^~-words of form w@?xw, Theoret. Comput. Sci. 393 (2008) 280-286] obtained p"n(k)=6 for all k@?N and n=2,3,4, and gave all C^b^@w-words of the form w@?xw with gap less than 5, which imply p"n(~)=2 for n=0,1, and p"n(~)=6 for n=2,3,4. In this paper, we prove the following intriguing results: (1) If w@?xw@?C^b^@w and |x|=7 then the first and last letters of the word x are the same; (2) p"n(~)=14 for n=5; (3) For every positive integer n, there exists a positive integer H(n) such that for all k@?N, if kH(n) then p"n(k)=p"5(k) if k is odd and p"n(k)=p"6(k) if k is even, which would help us understand better the complexity of finite C^~-words of the form w@?xw. Moreover, we provide all twenty eight C^b^@w-words of the form w@?xw.