The total run length of a word
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A run in a word is a periodic factor whose length is at least twice its period and which cannot be extended to the left or right (by a letter) to a factor with greater period. In recent years a great deal of work has been done on estimating the maximum number of runs that can occur in a word of length n. A number of associated problems have also been investigated. In this paper we consider a new variation on the theme. We say that the total run length (TRL) of a word is the sum of the lengths of the runs in the word and that τ(n) is the maximum TRL over all words of length n. We show that n2/8<τ(n)<47n2/72+2nn2/8<τ(n)<47n2/72+2n for all n. We also give a formula for the average total run length of words of length n over an alphabet of size α, and some other results.
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NOTICE: this is the author’s version of a work that was accepted for publication in Theoretical Computer Science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Theoretical Computer Science, Vol. 501(2013). DOI: 10.1016/j.tcs.2013.06.004
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