Abstract
We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider their empirical distribution obtained by an enumeration of binary cubefree words up to length 80.
Highlights
The interest in combinatorics on words goes back to the work of Axel Thue at the beginning of the20th century [1]
We review the recent progress on powerfree words, with emphasis on the two ‘classic’
We reviewed recent progress on the combinatorics of k-powerfree words, with particular emphasis on the examples of binary cubefree and ternary squarefree words, which have attracted most attention over the years
Summary
The interest in combinatorics on words goes back to the work of Axel Thue at the beginning of the20th century [1]. We proceed by introducing the entropy of k-powerfree words and summarise the methods to derive upper and lower bounds in general, and for binary cubefree and ternary squarefree words in particular. This limitation means that the number of forbidden words is finite, and that the resulting factorial set has a larger entropy than the set of k-powerfree words, so the latter provides an upper bound.
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