# Combinatorics on Words by M. Lothaire

By M. Lothaire

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Let w — axa2- • • an,n>l9 a^A. For K / < w , let \f/(i) be the greatest integer j < / — 1 such that axa2 • • «_,-_, = 0,-y+r • • at_l9 at with \f/(i) = 0 if no such integery exists. a. Show that the following algorithm computes \p: 2. i - l ; y « - 0 ; 3. while i 0 am/ ay ¥" dj do j 5. 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.

Show that the foregoing algorithm can be used to test whether a word uEA+ is a factor of a word vEA~*~. ) This is called a string-matching algorithm (see Knuth, Morris, and Pratt 1977). 1. Let F be the free group over the set A and H be a subgroup of F. a. Show that it is possible to -choose a set Q of representatives of the right cosets of H in F such that the set p(Q) of reduced words representing Q contains all its left factors. Such a set Q is called a Schreier system for H. b. Let Q be a Schreier system for H and X={paq\p9qeQ,aeA9paG(H-l)q}\ Show that X generates H.

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. By asumption, there are x, v, yE A*, cE A with /x(w) = xcvcvcy Note that |ct>ct>c| is odd, but [i(w)E X* with X— {ab, ba}: therefore |/x(w)| is even and \xy\ is odd. Thus • Either: | JC| is even, and x, cvcv, cyE X*, • Or: |JC| is odd, and JCC, vcvc, yE X*. This implies that | v \ is odd, since otherwise we get from cvcv E X* (resp.