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Báo cáo toán học: "On the Entropy and Letter Frequencies of Ternary Square-Free Words"
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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: On the Entropy and Letter Frequencies of Ternary Square-Free Words. | On the Entropy and Letter Frequencies of Ternary Square-Free Words Christoph Richard Institut fur Mathematik Universitat Greifswald Jahnstr. 15a 17487 Greifswald Germany richard@uni-greifswald.de Uwe Grimm Applied Mathematics Department The Open University Walton Hall Milton Keynes MK7 6AA UK u.g.grimm@open.ac.uk Submitted Mar 19 2003 Accepted Aug 28 2003 Published Feb 14 2004 Keywords Combinatorics on words square-free words Mathematics Subject Classifications 68R15 05A15 Abstract We enumerate ternary length- square-free words which are words avoiding squares of all words up to length I for I 24. We analyse the singular behaviour of the corresponding generating functions. This leads to new upper entropy bounds for ternary square-free words. We then consider ternary square-free words with fixed letter densities thereby proving exponential growth for certain ensembles with various letter densities. We derive consequences for the free energy and entropy of ternary square-free words. 1 Introduction The interest in the combinatorics of pattern-avoiding 3 2 8 in particular of power-free words goes back to work of Axel Thue in the early 20th century 37 38 . The celebrated Prouhet-Thue-Morse sequence defined by a substitution rule a ab and b ba on a two-letter alphabet a b proves the existence of infinite cube-free words in two letters a and b. Here a word of length n is a string of n letters from a certain alphabet E an element of the set En of n-letter words in E. The union E un 0 En is the language of all words in the alphabet E. It is a monoid with concatenation of words as operation and with the empty word A of zero length as neutral element 23 in particular E0 A . A word w is called square-free if w xyyz with words x y and z implies that y A is the empty word and cube-free words are defined analogously. So square-free words are characterised by the property that they do not contain an adjacent repetition of any subword. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 .