By Francine Blanchet-Sadri

The research of combinatorics on phrases is a comparatively new examine sector within the fields of discrete and algorithmic arithmetic. that includes an easy, obtainable variety, Algorithmic Combinatorics on Partial phrases offers combinatorial and algorithmic ideas within the rising box of phrases and partial phrases. This e-book incorporates a wealth of workouts and difficulties that assists with quite a few set of rules tracing, set of rules layout, mathematical proofs, and application implementation. additionally it is quite a few labored instance and diagrams, making this a worthwhile textual content for college kids, researchers, and practitioners trying to comprehend this complicated topic the place many difficulties stay unexplored.

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In general, to fully verify a given pword, gcd(k, l) sequences are needed corresponding to the positions 0 ≤ i < gcd(k, l). Now, we use these sequences 3 Some readers may find the following (equivalent) definition of seqk,l (i) more intuitive: i0 = i and for 1 ≤ j ≤ n + 1, ij = (ij−1 + k) mod (k + l). ) As before, continue the sequence until the first occurrence of i is reached. Combinatorial Properties of Partial Words 51 to make our definition of (k, l)-special partial word precise in the following manner.

3 Let a and b be letters in an alphabet A. Then, a6 = aaaaaa (aba)3 = (aba)(aba)(aba) = abaabaaba At this point, we define a word u to be primitive if there exists no word v such that u = v i with i ≥ 2. 4 The word u = abaaba is not primitive, as shown here: u = abaaba = (aba)2 = v 2 where v = aba 1A semigroup is a nonempty set together with a binary associative operation. A monoid is a semigroup with identity. Preliminaries on Partial Words 27 The word aaaaa = a5 is also clearly not primitive, whereas the word aaaab is primitive.

5 Let x and y be nonempty partial words such that xy has at most one hole. If xy ↑ yx, then there exists a word z such that x ⊂ z m and y ⊂ z n for some integers m, n. However, if xy possesses more than one hole, the situation becomes more subtle. Indeed, it is easy to produce a counterexample when xy contains just one more hole. 6 Let x = bb and y = abb . Then xy = bbabb ↑ abb bb = yx Since gcd(|x|, |y|) = 1, if x and y were contained in powers of a common word z, then |z| would be equal to 1, which is not possible for y.