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Abelian Square-Free Partial Words [chapter]

Francine Blanchet-Sadri, Jane I. Kim, Robert Mercaş, William Severa, Sean Simmons
2010 Lecture Notes in Computer Science  
Infinite abelian square-free words have been constructed over alphabets of sizes as small as four.  ...  In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes.  ...  The partial word w is abelian square-free if it does not have any full or partial abelian square, except those of the form a or a , where a ∈ A.  ... 
doi:10.1007/978-3-642-13089-2_8 fatcat:rq3duhwdwfefhdns37oivwckbq

On the number of partially abelian square-free words on a three-letter alphabet

Robert Cori, Maria Rosaria Formisano
1991 Theoretical Computer Science  
Let # be a &square-free morphism, consider any square-free word f of length not greater than 5; f is also &square-free, hence e(f) is square-free.  ...  El A morphism # from {a, b, c}* into itself is &square-free if, for any &square-free word f, e(f) is also &square-free.  ... 
doi:10.1016/0304-3975(91)90322-s fatcat:tehmhj3z4fe65cmmsz3ecdvhuy

Avoiding abelian squares in partial words

F. Blanchet-Sadri, Jane I. Kim, Robert Mercaş, William Severa, Sean Simmons, Dimin Xu
2012 Journal of combinatorial theory. Series A  
In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes.  ...  In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes.  ...  Robert Merca¸s' work was partially supported by Research Grant No. 1323 U07 E30 N-2008/InvAct/Bel, G.  ... 
doi:10.1016/j.jcta.2011.08.008 fatcat:5sarssevh5ao3iv2mcyriurudq

Abelian repetitions in partial words

F. Blanchet-Sadri, Sean Simmons, Dimin Xu
2012 Advances in Applied Mathematics  
Then we show, in a number of cases, that the number of abelian p-free partial words of length n with h holes over a given alphabet grows exponentially as n increases.  ...  We study abelian repetitions in partial words, or sequences that may contain some unknown positions or holes.  ...  The partial word u is abelian square-free if it does not have any full or partial abelian square, except those of the form a or a , where a ∈ A.  ... 
doi:10.1016/j.aam.2011.06.006 fatcat:cx2docar75hehmfhjxoaqnclym

Abelian Pattern Avoidance in Partial Words [chapter]

Francine Blanchet-Sadri, Sean Simmons
2012 Lecture Notes in Computer Science  
Recently, (abelian) pattern avoidance was initiated in the more general framework of partial words, which allow for undefined positions called holes.  ...  On the other hand, Erdős raised the question as to whether there exists an infinite word that avoids abelian squares, i.e., a word with no two adjacent factors being permutations of one another.  ...  A partial word is abelian square-free if it does not contain any factor that results in an abelian square after filling in the holes.  ... 
doi:10.1007/978-3-642-32589-2_21 fatcat:25ohmrc7lrgchgragwi5pje67y

Words with the Maximum Number of Abelian Squares [article]

Gabriele Fici, Filippo Mignosi
2015 arXiv   pre-print
An abelian square is the concatenation of two words that are anagrams of one another. A word of length n can contain Θ(n^2) distinct factors that are abelian squares.  ...  We study infinite words such that the number of abelian square factors of length n grows quadratically with n.  ...  We also proved that Sturmian words that are β-power free for some β ≥ 2 are uniformly abelian-square rich.  ... 
arXiv:1506.03562v1 fatcat:i2fgs6ze6zh6jfpuau3e5dpxqq

A powerful abelian square-free substitution over 4 letters

Veikko Keränen
2009 Theoretical Computer Science  
An abelian square is a non-empty word uv, where u and v are permutations (anagrams) of each other.  ...  In 1961, Paul Erdös posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet.  ...  Acknowledgements Originally, the author in [42] presented the main results of this paper at WORDS'07, the 6th International Conference on Combinatorics on Words, Institut de Mathématiques de Luminy,  ... 
doi:10.1016/j.tcs.2009.05.027 fatcat:p2orhjifxnfnzmy7wwc3i5lnw4

5-Abelian cubes are avoidable on binary alphabets

Robert Mercaş, Aleksi Saarela
2014 RAIRO - Theoretical Informatics and Applications  
For example, the word banana contains the squares anan and nana and the overlap anana. A  ...  For a word uvw, we say that uv is a square if u = v and both words are non-empty, and we say that uvw is an overlap if all words are non-empty, and either u = v and w is a prefix of u, or v = w and u is  ...  For example, intestines is an abelian square where intes is a permutation of the word tines. A word is abelian square-free, if it does not contain any abelian square as a factor.  ... 
doi:10.1051/ita/2014020 fatcat:mvooknvgpzhwxglv7w3gbukcc4

Pattern avoidance: themes and variations

James D. Currie
2005 Theoretical Computer Science  
We review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems.  ...  The study of abelian squares was initiated by Erdös [38] . As in the non-abelian case, the study of abelian square-free words has algebraic interpretations (see [46] ).  ...  Thus a square-free word avoids xx. More generally, a word which avoids An abelian square is a word x 1 x 2 where x 2 can be obtained from x 1 by rearranging letters.  ... 
doi:10.1016/j.tcs.2005.01.004 fatcat:cr5bmro63zbbnco3ogmsuw7yly

Abelian-square-rich words

Gabriele Fici, Filippo Mignosi, Jeffrey Shallit
2017 Theoretical Computer Science  
As for Sturmian words, we prove that a Sturmian word s_α of angle α is uniformly abelian-square-rich if and only if the irrational α has bounded partial quotients, that is, if and only if s_α has bounded  ...  An abelian square is the concatenation of two words that are anagrams of one another.  ...  Now we prove that if α has bounded partial quotients, then the Sturmian word s α is (uniformly) abelian-square-rich.  ... 
doi:10.1016/j.tcs.2017.02.012 fatcat:kvuemu6aw5djvkvyzv7hqghcce

On Abelian squares and substitutions

Arturo Carpi
1999 Theoretical Computer Science  
We study substitutions preserving Abelian square-free words and the more general notion of substitutions with bounded Abelian squares.  ...  In particular, we prove the existence of algorithms deciding whether a substitution mapping each letter into a set of commutatively equivalent words belongs to one of these classes.  ...  The Abelian square-free words exhibited in the quoted papers are obtained by iteration of morphisms preserving Abelian square-free words (Abelian square-free morphisms).  ... 
doi:10.1016/s0304-3975(98)00250-3 fatcat:tfxadtagmreblie2rimyuceij4

Avoidability of long k-abelian repetitions [article]

Michaël Rao, Matthieu Rosenfeld
2015 arXiv   pre-print
Then we show that one can avoid 3-abelian-squares of period at least 3 in infinite binary words and 2-abelian-squares of period at least 2 in infinite ternary words.  ...  Finally we study the minimum number of distinct k-abelian-squares that must appear in an infinite binary word.  ...  Then h 4 is (3, 3)-abelian-square-free. Moreover, for every abelian-square-free word w, h 4 (w) contains only 4 distinct 3-abelian-squares: 0 2 , 1 2 , (01) 2 and (10) 2 .  ... 
arXiv:1507.02581v1 fatcat:bvyju3mcivfmjl2iwk57gyrgdm

Square-Free Words over Partially Commutative Alphabets [chapter]

Łukasz Mikulski, Marcin Piątkowski, Wojciech Rytter
2015 Lecture Notes in Computer Science  
We give two characterizations (morphic and recursive) of the lexicographically minimal square-free word v in the case of a partially commutative alphabet of size three.  ...  We give two characterizations (morphic and recursive) of the lexicographically minimal square-free word v in the case of a partially commutative alphabet Θ of size three.  ...  Square-free and abelian square-free words have been extensively studied.  ... 
doi:10.1007/978-3-319-15579-1_33 fatcat:fqfho7swnrbilj4j3cpyhsczba

Partially abelian squarefree words

Robert Cori, Maria Rosaria Formisano
1990 RAIRO - Theoretical Informatics and Applications  
PARTIALLY ABELIAN SQUARE FREE WORDS IN A FOUR LETTERS ALPHABET In this section, A is the alphabet {a, b, c, d}.  ...  PARTIALLY ABELIAN SQUARE FREE WORDS IN { a, b, c }* In this section A is the alphabet consisting of the three letters {a, b, c} and 0 t is the relation consisting of the two pairs {{a, c), (c, a)}, 0 2  ... 
doi:10.1051/ita/1990240605091 fatcat:tyvyqptzvbdtxk4c56du3tqqmy

3-Abelian Cubes Are Avoidable on Binary Alphabets [chapter]

Robert Mercaş, Aleksi Saarela
2013 Lecture Notes in Computer Science  
A k-abelian cube is a word uvw, where u, v, w have the same factors of length at most k with the same multiplicities.  ...  Previously it has been known that k-abelian cubes are avoidable over a binary alphabet for k ≥ 5. Here it is proved that this holds for k ≥ 3.  ...  Computer experiments would suggest that there are 3-abelian square-free ternary words, but it is known that there are no pure morphic k-abelian square-free ternary words for any k [13] .  ... 
doi:10.1007/978-3-642-38771-5_33 fatcat:7povcv5jujghrjjhplhpvx3vs4
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