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A131747
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Number of morphically primitive words w in {1,2,3,...}^* of length n (modulo renaming), also the number of words (length n) which are fixed points of nontrivial morphisms, also the number of succinct patterns of length n in canonical form, also the number of words which have an unambiguous morphic image in {a,b}^*.
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0
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1, 1, 1, 3, 11, 32, 152, 625, 3152, 16154, 90993, 539181, 3398803, 22561791, 157493522, 1151863409, 8803760900, 70149141443, 581461668973, 5004153572628
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OFFSET
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1,4
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COMMENTS
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A word v is called morphically primitive iff there is no shorter word v' such that there exist a morphism mapping v onto v' and a morphism mapping v' onto v (the morphisms may erase symbols).
So far, no formula is known to calculate this number for any given n. Note that the number of all words w in {1,2,3,...}^* of length n modulo renaming corresponds to the Bell number of n (see A000110).
It would be interesting to know the limit of the ratio of morphically primitive words and all words (modulo renaming). For n = 15, this ratio is approx. 0.1139. Moreover, note that the terms "primitive (or aperiodic) words" and "morphically primitive words" are incomparable.
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REFERENCES
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Reidenbach, Daniel, and Johannes C. Schneider. "Morphically primitive words." (2009). See Table 1. Available from https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/4561/1/Reidenbach_Schneider_TCS_Morphically_primitive_words_Final_version.pdf [Do not delete this reference, because I do not know if the similar link below (which does not seem to work) refers to an identical version of the article. - N. J. A. Sloane, Jul 14 2018]
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LINKS
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D. Reidenbach and J. C. Schneider, Morphically Primitive Words. In Pierre Arnoux, Nicolas Bedaride and Julien Cassaigne, editors, Proc. 6th International Conference on Words, WORDS 2007, pages 262-272. 2007.
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EXAMPLE
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For length n = 4, the following words w in {1,2,3,...}^* are morphically primitive:
1 1 1 1
1 1 2 2
1 2 2 1
All other words of length 4 are either renamings of those words (e.g. 3 4 4 3 is a renaming of 1 2 2 1) or morphically imprimitive (e.g. 1 2 2 2).
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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Johannes C. Schneider (jschneider(AT)informatik.uni-kl.de), Oct 04 2007
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EXTENSIONS
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STATUS
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approved
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