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Cyclic Complexity of Some Infinite Words and Generalizations

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Author

Krawchuk, Colin
Rampersad, Narad

Uri

http://hdl.handle.net/10680/1417

Date

2018-03

Citation

Colin Krawchuk, Narad Rampersad, “Cyclic Complexity of Some Infinite Words and Generalizations,” Integers 18A (2018), #A12.

Abstract

Cassaigne et al. introduced the cyclic complexity function c_x(n), which gives the number of cyclic conjugacy classes of length-n factors of a word x. We study the behavior of this function for the Fibonacci word f and the Thue–Morse word t. If φ = (1 + √5)/2, we show that lim sup_{n → 1} c_f(n)/n ≥ 2/φ² and conjecture that equality holds. Similarly, we show that lim sup_{n → 1} c_t(n)/n ≥ 2 and conjecture that equality holds. We also propose a generalization of the cyclic complexity function and suggest some directions for further investigation. Most results are obtained by computer proofs using Mousavi’s Walnut software.

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  • Colin Krawchuk
  • Narad Rampersad

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