Algorithmic Combinatorics on Partial Words 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 innovations within the rising box of phrases and partial phrases. This ebook features a wealth of routines and difficulties that assists with various set of rules tracing, set of rules layout, mathematical proofs, and software implementation. it is usually a number of labored instance and diagrams, making this a invaluable textual content for college students, researchers, and practitioners trying to comprehend this advanced topic the place many difficulties stay unexplored.

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I) leads to y(i) = y(i + k) = · · · = y(i + (m − 1)k) = y(i + mk) = x(i + k − r) = · · · = x(i) since z is not {k, l}-special. Similarly, in the case where i ≥ r, seqk,l (i) = (i, i + k, . . , i + (m − 1)k, i + mk, i − r, . . , i) leads to y(i) = y(i+k) = · · · = y(i+(m−2)k) = y(i+(m−1)k) = x(i−r) = · · · = x(i). 5, the assumption of z being non {|x|, |y|}-special cannot be replaced by the weaker assumption of z not being (|x|, |y|)-special. 10. Here, z ⊂ xy and z ⊂ yx, but xy = yx. We end this chapter with the concept of a pairwise nonspecial set of partial words that is used in later chapters.

1 Alphabets, letters, and words Let A be a nonempty finite set of symbols, which we call an alphabet. An element a ∈ A is called a letter. A word over the alphabet A is a finite sequence of elements of A. 1 The following sets are alphabets: A = {a, b, c, n} B = {0, 1} The sequence of letters banana is a word over the alphabet A, as well as the word cbancb. Over the alphabet B, the sequences 0, 1, and 01010111110 are words. For any word u, α(u) is defined as the set of distinct letters in u. We allow for the possibility that a word consists of no letters.

Compute 1. The set of periods of u, P(u). 2. The set of weak periods of u, P (u). 3. 3 Let u and v be partial words. Prove that if v is primitive and v ⊂ u, then u is primitive as well. 4 S Let u be a partial word of length p, where p is a prime number. Prove that u is not primitive if and only if α(u) ≤ 1. 5 Construct a partial word with one hole of length 12 over the alphabet {a, b} that is weakly 5-periodic, weakly 8-periodic but not 1-periodic. 6 Let u be a word over an alphabet A, and let v = ua for any letter a in A.