Download PDF by M. Lothaire: Combinatorics on Words

By M. Lothaire

ISBN-10: 0521599245

ISBN-13: 9780521599245

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Example text

H , there exists a finite family {Sn'Jo)i}]^j^ri that covers {Snijo)i}l^J^r. K. bEK. ,/i}. ,^-}. Therefore S~ ' ^Eco, and thus d(S~n'Ja, b)

If | * | = 0 , then indeed aa,bb&X*. Let x E X*, x T^ 1 and suppose u — axa E X* (the case bxb E X* is similar). Then u — xxx2- - - xr, with * ! , . . , x r E X\ consequently xx — ab and xr — ba. Thus u — abyba with y — x2 • • • xr_ j E X*. But now by induction x = byb is not in X*, contrary to the assumption. 6. Let wE A +. If w has no overlapping factor, then fi(w) has no overlapping factor. LEMMA Proof Assume that /i(w>) has an overlapping factor for some wE A*. We show that w also has an overlapping factor.

I « - I + 1; y^-y + 1; 6. ) b. Show that the algorithm of problem part (a) can be used to test whether a word u is a factor of a word v. c. Show that the number of consecutive times the while loop of line 4 may be executed does not exceed the integer r such that where Ar is the rth term of the Fibonacci sequence. 6 that this bound can be reached. 1. 2 Let W c A + be a set of words such that no proper factor of a word of W is in W P = A*-A*WA* be the set of words having no factor in W. Let for each ue W, Xu = A*u-A*WA + be the set of words having u as a right factor but no other factor in W.

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Combinatorics on Words by M. Lothaire

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